Lecture 7: Random Variables and Probability Distributions API-201Z

Home , Bernoulli distribution, Bernoulli trial, Binomial distribution

Maya Sen

Harvard Kennedy School http://scholar.harvard.edu/msen I Problem Set #4 posted

I We’ll be posting several practice exams and practice problems in advance of midterm

Announcements I We’ll be posting several practice exams and practice problems in advance of midterm

Announcements

I Problem Set #4 posted Announcements

I Problem Set #4 posted

I We’ll be posting several practice exams and practice problems in advance of midterm I By now, be comfortable w/ summary statistics in Stata/R, basic probability, conditional probability, independence, LOTP, Bayes’ Rule

I Now:

Roadmap I Now:

Roadmap

I By now, be comfortable w/ summary statistics in Stata/R, basic probability, conditional probability, independence, LOTP, Bayes’ Rule I Introduce probability distributions, which are foundational for statistical inference I Random variables I Give examples of discrete and continuous random variables I Walk through probability distributions for discrete random variables (continuous next time) I Introduce Bernoulli processes

Roadmap

I By now, be comfortable w/ summary statistics in Stata/R, basic probability, conditional probability, independence, LOTP, Bayes’ Rule

I Now: I Random variables I Give examples of discrete and continuous random variables I Walk through probability distributions for discrete random variables (continuous next time) I Introduce Bernoulli processes

Roadmap

I By now, be comfortable w/ summary statistics in Stata/R, basic probability, conditional probability, independence, LOTP, Bayes’ Rule

I Now:

I Introduce probability distributions, which are foundational for statistical inference I Give examples of discrete and continuous random variables I Walk through probability distributions for discrete random variables (continuous next time) I Introduce Bernoulli processes

Roadmap

I By now, be comfortable w/ summary statistics in Stata/R, basic probability, conditional probability, independence, LOTP, Bayes’ Rule

I Now:

I Introduce probability distributions, which are foundational for statistical inference I Random variables I Walk through probability distributions for discrete random variables (continuous next time) I Introduce Bernoulli processes

Roadmap

I By now, be comfortable w/ summary statistics in Stata/R, basic probability, conditional probability, independence, LOTP, Bayes’ Rule

I Now:

I Introduce probability distributions, which are foundational for statistical inference I Random variables I Give examples of discrete and continuous random variables I Introduce Bernoulli processes

Roadmap

I By now, be comfortable w/ summary statistics in Stata/R, basic probability, conditional probability, independence, LOTP, Bayes’ Rule

I Now:

I By now, be comfortable w/ summary statistics in Stata/R, basic probability, conditional probability, independence, LOTP, Bayes’ Rule

I Now:

I Introduce probability distributions, which are foundational for statistical inference I Random variables I Give examples of discrete and continuous random variables I Walk through probability distributions for discrete random variables (continuous next time) I Introduce Bernoulli processes I Random phenomenon: A situation where we know what outcomes could happen, but we don’t know which particular outcome did or will happen

I Ex) Flipping a coin 3 times I Ex) Randomly calling 5 U.S. Senators I Ex) Lady Bristol choosing 4 cups of tea out of 8

I Random variables: (1) Numerical outcomes of (2) random phenomenon

I Ex) Number of Heads in 3 coin flips I Ex) Number of female Democrats you call I Ex) Number of 4 cups Lady identifies correctly I Can take on a set of possible different numerical values, each with an associated probability I RVs denoted by capital letters – e.g., X or Y I Note: Random variable = number (e.g, number of heads), random event = event (e.g., tossing heads)

I As before, “random” does not mean “haphazard”

What are random variables? : A situation where we know what outcomes could happen, but we don’t know which particular outcome did or will happen

I Ex) Flipping a coin 3 times I Ex) Randomly calling 5 U.S. Senators I Ex) Lady Bristol choosing 4 cups of tea out of 8

I Random variables: (1) Numerical outcomes of (2) random phenomenon

I As before, “random” does not mean “haphazard”

What are random variables?

I Random phenomenon I Ex) Flipping a coin 3 times I Ex) Randomly calling 5 U.S. Senators I Ex) Lady Bristol choosing 4 cups of tea out of 8

I Random variables: (1) Numerical outcomes of (2) random phenomenon

I As before, “random” does not mean “haphazard”

What are random variables?

I Random phenomenon: A situation where we know what outcomes could happen, but we don’t know which particular outcome did or will happen I Ex) Randomly calling 5 U.S. Senators I Ex) Lady Bristol choosing 4 cups of tea out of 8

I Random variables: (1) Numerical outcomes of (2) random phenomenon

I As before, “random” does not mean “haphazard”

What are random variables?

I Random phenomenon: A situation where we know what outcomes could happen, but we don’t know which particular outcome did or will happen

I Ex) Flipping a coin 3 times I Ex) Lady Bristol choosing 4 cups of tea out of 8

I Random variables: (1) Numerical outcomes of (2) random phenomenon

I As before, “random” does not mean “haphazard”

What are random variables?

I Random phenomenon: A situation where we know what outcomes could happen, but we don’t know which particular outcome did or will happen

I Ex) Flipping a coin 3 times I Ex) Randomly calling 5 U.S. Senators I Random variables: (1) Numerical outcomes of (2) random phenomenon

I As before, “random” does not mean “haphazard”

What are random variables?

I Random phenomenon: A situation where we know what outcomes could happen, but we don’t know which particular outcome did or will happen

I Ex) Flipping a coin 3 times I Ex) Randomly calling 5 U.S. Senators I Ex) Lady Bristol choosing 4 cups of tea out of 8 : (1) Numerical outcomes of (2) random phenomenon

I As before, “random” does not mean “haphazard”

What are random variables?

I Random phenomenon: A situation where we know what outcomes could happen, but we don’t know which particular outcome did or will happen

I Ex) Flipping a coin 3 times I Ex) Randomly calling 5 U.S. Senators I Ex) Lady Bristol choosing 4 cups of tea out of 8

I Random variables I Ex) Number of Heads in 3 coin flips I Ex) Number of female Democrats you call I Ex) Number of 4 cups Lady identifies correctly I Can take on a set of possible different numerical values, each with an associated probability I RVs denoted by capital letters – e.g., X or Y I Note: Random variable = number (e.g, number of heads), random event = event (e.g., tossing heads)

I As before, “random” does not mean “haphazard”

What are random variables?

I Random phenomenon: A situation where we know what outcomes could happen, but we don’t know which particular outcome did or will happen

I Ex) Flipping a coin 3 times I Ex) Randomly calling 5 U.S. Senators I Ex) Lady Bristol choosing 4 cups of tea out of 8

I Random variables: (1) Numerical outcomes of (2) random phenomenon I Ex) Number of female Democrats you call I Ex) Number of 4 cups Lady identiﬁes correctly I Can take on a set of possible diﬀerent numerical values, each with an associated probability I RVs denoted by capital letters – e.g., X or Y I Note: Random variable = number (e.g, number of heads), random event = event (e.g., tossing heads)

I As before, “random” does not mean “haphazard”

What are random variables?

I Random phenomenon: A situation where we know what outcomes could happen, but we don’t know which particular outcome did or will happen

I Ex) Flipping a coin 3 times I Ex) Randomly calling 5 U.S. Senators I Ex) Lady Bristol choosing 4 cups of tea out of 8

I Random variables: (1) Numerical outcomes of (2) random phenomenon

I Ex) Number of Heads in 3 coin flips I Ex) Number of 4 cups Lady identifies correctly I Can take on a set of possible different numerical values, each with an associated probability I RVs denoted by capital letters – e.g., X or Y I Note: Random variable = number (e.g, number of heads), random event = event (e.g., tossing heads)

I As before, “random” does not mean “haphazard”

What are random variables?

I Random phenomenon: A situation where we know what outcomes could happen, but we don’t know which particular outcome did or will happen

I Ex) Flipping a coin 3 times I Ex) Randomly calling 5 U.S. Senators I Ex) Lady Bristol choosing 4 cups of tea out of 8

I Random variables: (1) Numerical outcomes of (2) random phenomenon

I Ex) Number of Heads in 3 coin ﬂips I Ex) Number of female Democrats you call I Can take on a set of possible diﬀerent numerical values, each with an associated probability I RVs denoted by capital letters – e.g., X or Y I Note: Random variable = number (e.g, number of heads), random event = event (e.g., tossing heads)

I As before, “random” does not mean “haphazard”

What are random variables?

I Random phenomenon: A situation where we know what outcomes could happen, but we don’t know which particular outcome did or will happen

I Ex) Flipping a coin 3 times I Ex) Randomly calling 5 U.S. Senators I Ex) Lady Bristol choosing 4 cups of tea out of 8

I Random variables: (1) Numerical outcomes of (2) random phenomenon

I Ex) Number of Heads in 3 coin ﬂips I Ex) Number of female Democrats you call I Ex) Number of 4 cups Lady identiﬁes correctly I RVs denoted by capital letters – e.g., X or Y I Note: Random variable = number (e.g, number of heads), random event = event (e.g., tossing heads)

I As before, “random” does not mean “haphazard”

What are random variables?

I Random phenomenon: A situation where we know what outcomes could happen, but we don’t know which particular outcome did or will happen

I Ex) Flipping a coin 3 times I Ex) Randomly calling 5 U.S. Senators I Ex) Lady Bristol choosing 4 cups of tea out of 8

I Random variables: (1) Numerical outcomes of (2) random phenomenon

I Ex) Number of Heads in 3 coin flips I Ex) Number of female Democrats you call I Ex) Number of 4 cups Lady identifies correctly I Can take on a set of possible different numerical values, each with an associated probability I Note: Random variable = number (e.g, number of heads), random event = event (e.g., tossing heads)

I As before, “random” does not mean “haphazard”

What are random variables?

I Random phenomenon: A situation where we know what outcomes could happen, but we don’t know which particular outcome did or will happen

I Ex) Flipping a coin 3 times I Ex) Randomly calling 5 U.S. Senators I Ex) Lady Bristol choosing 4 cups of tea out of 8

I Random variables: (1) Numerical outcomes of (2) random phenomenon

I Ex) Number of Heads in 3 coin flips I Ex) Number of female Democrats you call I Ex) Number of 4 cups Lady identifies correctly I Can take on a set of possible different numerical values, each with an associated probability I RVs denoted by capital letters – e.g., X or Y I As before, “random” does not mean “haphazard”

What are random variables?

I Random phenomenon: A situation where we know what outcomes could happen, but we don’t know which particular outcome did or will happen

I Ex) Flipping a coin 3 times I Ex) Randomly calling 5 U.S. Senators I Ex) Lady Bristol choosing 4 cups of tea out of 8

I Random variables: (1) Numerical outcomes of (2) random phenomenon

I Random phenomenon: A situation where we know what outcomes could happen, but we don’t know which particular outcome did or will happen

I Ex) Flipping a coin 3 times I Ex) Randomly calling 5 U.S. Senators I Ex) Lady Bristol choosing 4 cups of tea out of 8

I Random variables: (1) Numerical outcomes of (2) random phenomenon

I As before, “random” does not mean “haphazard” I Example: You toss a coin 3 times

I Random phenomenon → act of tossing the coin 3 times I Random variable →

I Sequence of outcomes of the ﬂips, e.g. HTH, is not a random variable (not a number) I But we could make it into a random variable, X , by making it # of Heads in 3 ﬂips

I Note: Sample space can have many different random variables defined on it (e.g., number of tails flips, Y )

I Random variable → one of several possible mapping from sample space (HTH) into a number (in this case, 2)

What are random variables? I Random phenomenon → act of tossing the coin 3 times I Random variable →

I Sequence of outcomes of the ﬂips, e.g. HTH, is not a random variable (not a number) I But we could make it into a random variable, X , by making it # of Heads in 3 ﬂips

I Note: Sample space can have many different random variables defined on it (e.g., number of tails flips, Y )

I Random variable → one of several possible mapping from sample space (HTH) into a number (in this case, 2)

What are random variables?

I Example: You toss a coin 3 times I Random variable →

I Sequence of outcomes of the ﬂips, e.g. HTH, is not a random variable (not a number) I But we could make it into a random variable, X , by making it # of Heads in 3 ﬂips

I Note: Sample space can have many different random variables defined on it (e.g., number of tails flips, Y )

I Random variable → one of several possible mapping from sample space (HTH) into a number (in this case, 2)

What are random variables?

I Example: You toss a coin 3 times

I Random phenomenon → act of tossing the coin 3 times I Sequence of outcomes of the ﬂips, e.g. HTH, is not a random variable (not a number) I But we could make it into a random variable, X , by making it # of Heads in 3 ﬂips

I Note: Sample space can have many different random variables defined on it (e.g., number of tails flips, Y )

I Random variable → one of several possible mapping from sample space (HTH) into a number (in this case, 2)

What are random variables?

I Example: You toss a coin 3 times

I Random phenomenon → act of tossing the coin 3 times I Random variable → I But we could make it into a random variable, X , by making it # of Heads in 3 ﬂips

I Note: Sample space can have many different random variables defined on it (e.g., number of tails flips, Y )

I Random variable → one of several possible mapping from sample space (HTH) into a number (in this case, 2)

What are random variables?

I Example: You toss a coin 3 times

I Random phenomenon → act of tossing the coin 3 times I Random variable →

I Sequence of outcomes of the flips, e.g. HTH, is not a random variable (not a number) I Note: Sample space can have many different random variables defined on it (e.g., number of tails flips, Y )

I Random variable → one of several possible mapping from sample space (HTH) into a number (in this case, 2)

What are random variables?

I Example: You toss a coin 3 times

I Random phenomenon → act of tossing the coin 3 times I Random variable →

I Sequence of outcomes of the ﬂips, e.g. HTH, is not a random variable (not a number) I But we could make it into a random variable, X , by making it # of Heads in 3 ﬂips I Random variable → one of several possible mapping from sample space (HTH) into a number (in this case, 2)

What are random variables?

I Example: You toss a coin 3 times

I Random phenomenon → act of tossing the coin 3 times I Random variable →

I Sequence of outcomes of the ﬂips, e.g. HTH, is not a random variable (not a number) I But we could make it into a random variable, X , by making it # of Heads in 3 ﬂips

I Note: Sample space can have many different random variables defined on it (e.g., number of tails flips, Y ) What are random variables?

I Example: You toss a coin 3 times

I Random phenomenon → act of tossing the coin 3 times I Random variable →

I Sequence of outcomes of the ﬂips, e.g. HTH, is not a random variable (not a number) I But we could make it into a random variable, X , by making it # of Heads in 3 ﬂips

I Note: Sample space can have many different random variables defined on it (e.g., number of tails flips, Y )

I Random variable → one of several possible mapping from sample space (HTH) into a number (in this case, 2) I A discrete random variable can take on only integer (countable) number of values (usually within an interval)

I Examples:

I Outcomes when you roll a roulette wheel I Number of times a particular word is used in a document I Number of Democrats in U.S. Senate

I → In theory, you could count up these values, given ﬁnite amount of time

Discrete Random Variables I Examples:

I Outcomes when you roll a roulette wheel I Number of times a particular word is used in a document I Number of Democrats in U.S. Senate

I → In theory, you could count up these values, given ﬁnite amount of time

Discrete Random Variables

I A discrete random variable can take on only integer (countable) number of values (usually within an interval) I Outcomes when you roll a roulette wheel I Number of times a particular word is used in a document I Number of Democrats in U.S. Senate

I → In theory, you could count up these values, given ﬁnite amount of time

Discrete Random Variables

I A discrete random variable can take on only integer (countable) number of values (usually within an interval)

I Examples: I Number of times a particular word is used in a document I Number of Democrats in U.S. Senate

I → In theory, you could count up these values, given ﬁnite amount of time

Discrete Random Variables

I A discrete random variable can take on only integer (countable) number of values (usually within an interval)

I Examples:

I Outcomes when you roll a roulette wheel I Number of Democrats in U.S. Senate

I → In theory, you could count up these values, given ﬁnite amount of time

Discrete Random Variables

I A discrete random variable can take on only integer (countable) number of values (usually within an interval)

I Examples:

I Outcomes when you roll a roulette wheel I Number of times a particular word is used in a document I → In theory, you could count up these values, given ﬁnite amount of time

Discrete Random Variables

I A discrete random variable can take on only integer (countable) number of values (usually within an interval)

I Examples:

I Outcomes when you roll a roulette wheel I Number of times a particular word is used in a document I Number of Democrats in U.S. Senate Discrete Random Variables

I A discrete random variable can take on only integer (countable) number of values (usually within an interval)

I Examples:

I Outcomes when you roll a roulette wheel I Number of times a particular word is used in a document I Number of Democrats in U.S. Senate

I → In theory, you could count up these values, given ﬁnite amount of time I A continuous random variable that can take on any value (usually within an interval) on the real number line

I Examples:

I Annual rainfall in Bangladesh I Time it takes to run a 100m dash I Time until next earthquake in Japan

I → Cannot count these quantities, given ﬁnite amount of time

I Much of intuition we cover today applies to both

Continuous Random Variables (next time) I Examples:

I Annual rainfall in Bangladesh I Time it takes to run a 100m dash I Time until next earthquake in Japan

I → Cannot count these quantities, given ﬁnite amount of time

I Much of intuition we cover today applies to both

Continuous Random Variables (next time)

I A continuous random variable that can take on any value (usually within an interval) on the real number line I Annual rainfall in Bangladesh I Time it takes to run a 100m dash I Time until next earthquake in Japan

I → Cannot count these quantities, given ﬁnite amount of time

I Much of intuition we cover today applies to both

Continuous Random Variables (next time)

I A continuous random variable that can take on any value (usually within an interval) on the real number line

I Examples: I Time it takes to run a 100m dash I Time until next earthquake in Japan

I → Cannot count these quantities, given ﬁnite amount of time

I Much of intuition we cover today applies to both

Continuous Random Variables (next time)

I A continuous random variable that can take on any value (usually within an interval) on the real number line

I Examples:

I Annual rainfall in Bangladesh I Time until next earthquake in Japan

I → Cannot count these quantities, given ﬁnite amount of time

I Much of intuition we cover today applies to both

Continuous Random Variables (next time)

I A continuous random variable that can take on any value (usually within an interval) on the real number line

I Examples:

I Annual rainfall in Bangladesh I Time it takes to run a 100m dash I → Cannot count these quantities, given ﬁnite amount of time

I Much of intuition we cover today applies to both

Continuous Random Variables (next time)

I A continuous random variable that can take on any value (usually within an interval) on the real number line

I Examples:

I Annual rainfall in Bangladesh I Time it takes to run a 100m dash I Time until next earthquake in Japan I Much of intuition we cover today applies to both

Continuous Random Variables (next time)

I A continuous random variable that can take on any value (usually within an interval) on the real number line

I Examples:

I Annual rainfall in Bangladesh I Time it takes to run a 100m dash I Time until next earthquake in Japan

I → Cannot count these quantities, given ﬁnite amount of time Continuous Random Variables (next time)

I A continuous random variable that can take on any value (usually within an interval) on the real number line

I Examples:

I Annual rainfall in Bangladesh I Time it takes to run a 100m dash I Time until next earthquake in Japan

I → Cannot count these quantities, given ﬁnite amount of time

I Much of intuition we cover today applies to both I We describe distribution of random variables using probability distributions

I Probability distributions represent information about how likely various outcomes of X are

I 2 common ways of representing probability distributions for discrete random variables are via: 1. Probability mass functions( f (x)) 2. Cumulative mass functions( F (x))

I Similar for continuous distributions: probability density functions, cumulative density functions

How do we describe random variables? I Probability distributions represent information about how likely various outcomes of X are

I 2 common ways of representing probability distributions for discrete random variables are via: 1. Probability mass functions( f (x)) 2. Cumulative mass functions( F (x))

I Similar for continuous distributions: probability density functions, cumulative density functions

How do we describe random variables?

I We describe distribution of random variables using probability distributions I 2 common ways of representing probability distributions for discrete random variables are via: 1. Probability mass functions( f (x)) 2. Cumulative mass functions( F (x))

I Similar for continuous distributions: probability density functions, cumulative density functions

How do we describe random variables?

I We describe distribution of random variables using probability distributions

I Probability distributions represent information about how likely various outcomes of X are 1. Probability mass functions( f (x)) 2. Cumulative mass functions( F (x))

I Similar for continuous distributions: probability density functions, cumulative density functions

How do we describe random variables?

I We describe distribution of random variables using probability distributions

I Probability distributions represent information about how likely various outcomes of X are

I 2 common ways of representing probability distributions for discrete random variables are via: 2. Cumulative mass functions( F (x))

I Similar for continuous distributions: probability density functions, cumulative density functions

How do we describe random variables?

I We describe distribution of random variables using probability distributions

I Probability distributions represent information about how likely various outcomes of X are

I 2 common ways of representing probability distributions for discrete random variables are via: 1. Probability mass functions( f (x)) I Similar for continuous distributions: probability density functions, cumulative density functions

How do we describe random variables?

I We describe distribution of random variables using probability distributions

I Probability distributions represent information about how likely various outcomes of X are

I 2 common ways of representing probability distributions for discrete random variables are via: 1. Probability mass functions( f (x)) 2. Cumulative mass functions( F (x)) How do we describe random variables?

I We describe distribution of random variables using probability distributions

I Probability distributions represent information about how likely various outcomes of X are

I 2 common ways of representing probability distributions for discrete random variables are via: 1. Probability mass functions( f (x)) 2. Cumulative mass functions( F (x))

I Similar for continuous distributions: probability density functions, cumulative density functions I Probability mass functions: A function that deﬁnes the probability of each possible outcome

f (x) = P(X = x)

where X is the random variable, x possible value it could take

I Provides you with all possible outcomes and probability of each outcome

I Three ways of describing PMFs

I Table I Relative frequency histogram (bar plot) I Formula itself

Probability mass functions (PMFs) f (x) = P(X = x)

where X is the random variable, x possible value it could take

I Provides you with all possible outcomes and probability of each outcome

I Three ways of describing PMFs

I Table I Relative frequency histogram (bar plot) I Formula itself

Probability mass functions (PMFs)

I Probability mass functions: A function that deﬁnes the probability of each possible outcome where X is the random variable, x possible value it could take

I Provides you with all possible outcomes and probability of each outcome

I Three ways of describing PMFs

I Table I Relative frequency histogram (bar plot) I Formula itself

Probability mass functions (PMFs)

I Probability mass functions: A function that deﬁnes the probability of each possible outcome

f (x) = P(X = x) I Provides you with all possible outcomes and probability of each outcome

I Three ways of describing PMFs

I Table I Relative frequency histogram (bar plot) I Formula itself

Probability mass functions (PMFs)

I Probability mass functions: A function that deﬁnes the probability of each possible outcome

f (x) = P(X = x)

where X is the random variable, x possible value it could take I Three ways of describing PMFs

I Table I Relative frequency histogram (bar plot) I Formula itself

Probability mass functions (PMFs)

I Probability mass functions: A function that deﬁnes the probability of each possible outcome

f (x) = P(X = x)

where X is the random variable, x possible value it could take

I Provides you with all possible outcomes and probability of each outcome I Table I Relative frequency histogram (bar plot) I Formula itself

Probability mass functions (PMFs)

I Probability mass functions: A function that deﬁnes the probability of each possible outcome

f (x) = P(X = x)

where X is the random variable, x possible value it could take

I Provides you with all possible outcomes and probability of each outcome

I Three ways of describing PMFs I Relative frequency histogram (bar plot) I Formula itself

Probability mass functions (PMFs)

I Probability mass functions: A function that deﬁnes the probability of each possible outcome

f (x) = P(X = x)

where X is the random variable, x possible value it could take

I Provides you with all possible outcomes and probability of each outcome

I Three ways of describing PMFs

I Table I Formula itself

Probability mass functions (PMFs)

I Probability mass functions: A function that deﬁnes the probability of each possible outcome

f (x) = P(X = x)

where X is the random variable, x possible value it could take

I Provides you with all possible outcomes and probability of each outcome

I Three ways of describing PMFs

I Table I Relative frequency histogram (bar plot) Probability mass functions (PMFs)

I Probability mass functions: A function that deﬁnes the probability of each possible outcome

f (x) = P(X = x)

where X is the random variable, x possible value it could take

I Provides you with all possible outcomes and probability of each outcome

I Three ways of describing PMFs

I Table I Relative frequency histogram (bar plot) I Formula itself I Let X be the number of heads in 3 coin ﬂips

I Sample space of possible outcomes would look like:

Event Value of X 1) TTT 0 2) TTH 1 3) THT 1 4) HTT 1 5) THH 2 6) HTH 2 7) HHT 2 8) HHH 3

Discrete Example I Sample space of possible outcomes would look like:

Event Value of X 1) TTT 0 2) TTH 1 3) THT 1 4) HTT 1 5) THH 2 6) HTH 2 7) HHT 2 8) HHH 3

Discrete Example

I Let X be the number of heads in 3 coin ﬂips Event Value of X 1) TTT 0 2) TTH 1 3) THT 1 4) HTT 1 5) THH 2 6) HTH 2 7) HHT 2 8) HHH 3

Discrete Example

I Let X be the number of heads in 3 coin ﬂips

I Sample space of possible outcomes would look like: 1) TTT 0 2) TTH 1 3) THT 1 4) HTT 1 5) THH 2 6) HTH 2 7) HHT 2 8) HHH 3

Discrete Example

I Let X be the number of heads in 3 coin ﬂips

I Sample space of possible outcomes would look like:

Event Value of X 2) TTH 1 3) THT 1 4) HTT 1 5) THH 2 6) HTH 2 7) HHT 2 8) HHH 3

Discrete Example

I Let X be the number of heads in 3 coin ﬂips

I Sample space of possible outcomes would look like:

Event Value of X 1) TTT 0 Discrete Example

I Let X be the number of heads in 3 coin ﬂips

I Sample space of possible outcomes would look like:

Event Value of X 1) TTT 0 2) TTH 1 3) THT 1 4) HTT 1 5) THH 2 6) HTH 2 7) HHT 2 8) HHH 3 I Let X be the number of heads in 3 coin ﬂips

I Probability mass function distribution would look like 0 1 2 3 f (x) 1/8 3/8 3/8 1/8

I All probabilities, between 0 and 1, inclusive

I Probabilities add up to 1

I Note: You’d only know this distribution if you repeated the random phenomenon over and over again!

Discrete Example I Probability mass function distribution would look like 0 1 2 3 f (x) 1/8 3/8 3/8 1/8

I All probabilities, between 0 and 1, inclusive

I Probabilities add up to 1

I Note: You’d only know this distribution if you repeated the random phenomenon over and over again!

Discrete Example

I Let X be the number of heads in 3 coin ﬂips 0 1 2 3 f (x) 1/8 3/8 3/8 1/8

I All probabilities, between 0 and 1, inclusive

I Probabilities add up to 1

I Note: You’d only know this distribution if you repeated the random phenomenon over and over again!

Discrete Example

I Let X be the number of heads in 3 coin ﬂips

I Probability mass function distribution would look like f (x) 1/8 3/8 3/8 1/8

I All probabilities, between 0 and 1, inclusive

I Probabilities add up to 1

I Note: You’d only know this distribution if you repeated the random phenomenon over and over again!

Discrete Example

I Let X be the number of heads in 3 coin ﬂips

I Probability mass function distribution would look like 0 1 2 3 I All probabilities, between 0 and 1, inclusive

I Probabilities add up to 1

I Note: You’d only know this distribution if you repeated the random phenomenon over and over again!

Discrete Example

I Let X be the number of heads in 3 coin ﬂips

I Probability mass function distribution would look like 0 1 2 3 f (x) 1/8 3/8 3/8 1/8 I Probabilities add up to 1

I Note: You’d only know this distribution if you repeated the random phenomenon over and over again!

Discrete Example

I Let X be the number of heads in 3 coin ﬂips

I Probability mass function distribution would look like 0 1 2 3 f (x) 1/8 3/8 3/8 1/8

I All probabilities, between 0 and 1, inclusive I Note: You’d only know this distribution if you repeated the random phenomenon over and over again!

Discrete Example

I Let X be the number of heads in 3 coin ﬂips

I Probability mass function distribution would look like 0 1 2 3 f (x) 1/8 3/8 3/8 1/8

I All probabilities, between 0 and 1, inclusive

I Probabilities add up to 1 Discrete Example

I Let X be the number of heads in 3 coin ﬂips

I Probability mass function distribution would look like 0 1 2 3 f (x) 1/8 3/8 3/8 1/8

I All probabilities, between 0 and 1, inclusive

I Probabilities add up to 1

I Note: You’d only know this distribution if you repeated the random phenomenon over and over again! I Could also represent this graphically:

Probability Mass Distribution 0.5 0.4 0.3 0.2 0.1 0.0 X = 0 X = 1 X = 2 X = 3

I Again note: Heights of bars add up to 1

Discrete Example Probability Mass Distribution 0.5 0.4 0.3 0.2 0.1 0.0 X = 0 X = 1 X = 2 X = 3

I Again note: Heights of bars add up to 1

Discrete Example

I Could also represent this graphically: I Again note: Heights of bars add up to 1

Discrete Example

I Could also represent this graphically:

Probability Mass Distribution 0.5 0.4 0.3 0.2 0.1 0.0 X = 0 X = 1 X = 2 X = 3 Discrete Example

I Could also represent this graphically:

Probability Mass Distribution 0.5 0.4 0.3 0.2 0.1 0.0 X = 0 X = 1 X = 2 X = 3

I Again note: Heights of bars add up to 1 I Probability mass function describes the probability of each possible outcome, f (x) = P(X = x)

I A cumulative mass function (CMF) gives us the probability that a random variable X takes on a value less than or equal to some particular value x:

F (x) = P(X 6 x) = P(X = x) X x X6

I Can also represent CMFs using:

I Table I Relative frequency histogram I Formula itself

Cumulative Mass Function I A cumulative mass function (CMF) gives us the probability that a random variable X takes on a value less than or equal to some particular value x:

F (x) = P(X 6 x) = P(X = x) X x X6

I Can also represent CMFs using:

I Table I Relative frequency histogram I Formula itself

Cumulative Mass Function

I Probability mass function describes the probability of each possible outcome, f (x) = P(X = x) F (x) = P(X 6 x) = P(X = x) X x X6

I Can also represent CMFs using:

I Table I Relative frequency histogram I Formula itself

Cumulative Mass Function

I Probability mass function describes the probability of each possible outcome, f (x) = P(X = x)

I A cumulative mass function (CMF) gives us the probability that a random variable X takes on a value less than or equal to some particular value x: I Can also represent CMFs using:

I Table I Relative frequency histogram I Formula itself

Cumulative Mass Function

I Probability mass function describes the probability of each possible outcome, f (x) = P(X = x)

I A cumulative mass function (CMF) gives us the probability that a random variable X takes on a value less than or equal to some particular value x:

F (x) = P(X 6 x) = P(X = x) X x X6 I Table I Relative frequency histogram I Formula itself

Cumulative Mass Function

I Probability mass function describes the probability of each possible outcome, f (x) = P(X = x)

I A cumulative mass function (CMF) gives us the probability that a random variable X takes on a value less than or equal to some particular value x:

F (x) = P(X 6 x) = P(X = x) X x X6

I Can also represent CMFs using: I Relative frequency histogram I Formula itself

Cumulative Mass Function

I Probability mass function describes the probability of each possible outcome, f (x) = P(X = x)

I A cumulative mass function (CMF) gives us the probability that a random variable X takes on a value less than or equal to some particular value x:

F (x) = P(X 6 x) = P(X = x) X x X6

I Can also represent CMFs using:

I Table I Formula itself

Cumulative Mass Function

I Probability mass function describes the probability of each possible outcome, f (x) = P(X = x)

I A cumulative mass function (CMF) gives us the probability that a random variable X takes on a value less than or equal to some particular value x:

F (x) = P(X 6 x) = P(X = x) X x X6

I Can also represent CMFs using:

I Table I Relative frequency histogram Cumulative Mass Function

I Probability mass function describes the probability of each possible outcome, f (x) = P(X = x)

I A cumulative mass function (CMF) gives us the probability that a random variable X takes on a value less than or equal to some particular value x:

F (x) = P(X 6 x) = P(X = x) X x X6

I Can also represent CMFs using:

I Table I Relative frequency histogram I Formula itself I Let X be the number of heads in 3 coin ﬂips

I Universe of possible outcomes would look like:

Event Value of X 1) TTT 0 2) TTH 1 3) THT 1 4) HTT 1 5) THH 2 6) HTH 2 7) HHT 2 8) HHH 3

Discrete Example I Universe of possible outcomes would look like:

Event Value of X 1) TTT 0 2) TTH 1 3) THT 1 4) HTT 1 5) THH 2 6) HTH 2 7) HHT 2 8) HHH 3

Discrete Example

I Let X be the number of heads in 3 coin ﬂips Event Value of X 1) TTT 0 2) TTH 1 3) THT 1 4) HTT 1 5) THH 2 6) HTH 2 7) HHT 2 8) HHH 3

Discrete Example

I Let X be the number of heads in 3 coin ﬂips

I Universe of possible outcomes would look like: Discrete Example

I Let X be the number of heads in 3 coin ﬂips

I Universe of possible outcomes would look like:

Event Value of X 1) TTT 0 2) TTH 1 3) THT 1 4) HTT 1 5) THH 2 6) HTH 2 7) HHT 2 8) HHH 3 I Let X be the number of heads in 3 coin ﬂips

I Probability mass function distribution would look like 0 1 2 3 f (x) 1/8 3/8 3/8 1/8

I Cumulative mass function distribution would look like: 0 1 2 3 F (x) 1/8 4/8 7/8 1

Discrete Example I Probability mass function distribution would look like 0 1 2 3 f (x) 1/8 3/8 3/8 1/8

I Cumulative mass function distribution would look like: 0 1 2 3 F (x) 1/8 4/8 7/8 1

Discrete Example

I Let X be the number of heads in 3 coin ﬂips 0 1 2 3 f (x) 1/8 3/8 3/8 1/8

I Cumulative mass function distribution would look like: 0 1 2 3 F (x) 1/8 4/8 7/8 1

Discrete Example

I Let X be the number of heads in 3 coin ﬂips

I Probability mass function distribution would look like I Cumulative mass function distribution would look like: 0 1 2 3 F (x) 1/8 4/8 7/8 1

Discrete Example

I Let X be the number of heads in 3 coin ﬂips

I Probability mass function distribution would look like 0 1 2 3 f (x) 1/8 3/8 3/8 1/8 0 1 2 3 F (x) 1/8 4/8 7/8 1

Discrete Example

I Let X be the number of heads in 3 coin ﬂips

I Probability mass function distribution would look like 0 1 2 3 f (x) 1/8 3/8 3/8 1/8

I Cumulative mass function distribution would look like: Discrete Example

I Let X be the number of heads in 3 coin ﬂips

I Probability mass function distribution would look like 0 1 2 3 f (x) 1/8 3/8 3/8 1/8

I Cumulative mass function distribution would look like: 0 1 2 3 F (x) 1/8 4/8 7/8 1 I And then we can also represent the CMF graphically:

Probability Mass Distribution Cumulative Mass Distribution 0.5 1.0 0.4 0.8 0.3 0.6 0.2 0.4 0.1 0.2 0.0 0.0 X = 0 X = 1 X = 2 X = 3 X ≤ 0 X ≤ 1 X ≤ 2 X ≤ 3

Discrete Example Probability Mass Distribution Cumulative Mass Distribution 0.5 1.0 0.4 0.8 0.3 0.6 0.2 0.4 0.1 0.2 0.0 0.0 X = 0 X = 1 X = 2 X = 3 X ≤ 0 X ≤ 1 X ≤ 2 X ≤ 3

Discrete Example

I And then we can also represent the CMF graphically: Discrete Example

I And then we can also represent the CMF graphically:

Probability Mass Distribution Cumulative Mass Distribution 0.5 1.0 0.4 0.8 0.3 0.6 0.2 0.4 0.1 0.2 0.0 0.0 X = 0 X = 1 X = 2 X = 3 X ≤ 0 X ≤ 1 X ≤ 2 X ≤ 3 I There are 5 candidates for 2 job openings I 3 of the candidates are women and 2 are men:

I W1, W2, W3, M1, M2

I Let X be a random variable that is the # of women hired

I X ∈ {0, 1, 2}

I What is the PMF of X ?

I What is the CMF of X ?

Sex Discrimination Example I 3 of the candidates are women and 2 are men:

I W1, W2, W3, M1, M2

I Let X be a random variable that is the # of women hired

I X ∈ {0, 1, 2}

I What is the PMF of X ?

I What is the CMF of X ?

Sex Discrimination Example

I There are 5 candidates for 2 job openings I W1, W2, W3, M1, M2

I Let X be a random variable that is the # of women hired

I X ∈ {0, 1, 2}

I What is the PMF of X ?

I What is the CMF of X ?

Sex Discrimination Example

I There are 5 candidates for 2 job openings I 3 of the candidates are women and 2 are men: I W1, W2, W3, M1, M2

I Let X be a random variable that is the # of women hired

I X ∈ {0, 1, 2}

I What is the PMF of X ?

I What is the CMF of X ?

Sex Discrimination Example

I There are 5 candidates for 2 job openings I 3 of the candidates are women and 2 are men: I Let X be a random variable that is the # of women hired

I X ∈ {0, 1, 2}

I What is the PMF of X ?

I What is the CMF of X ?

Sex Discrimination Example

I There are 5 candidates for 2 job openings I 3 of the candidates are women and 2 are men:

I W1, W2, W3, M1, M2 I X ∈ {0, 1, 2}

I What is the PMF of X ?

I What is the CMF of X ?

Sex Discrimination Example

I There are 5 candidates for 2 job openings I 3 of the candidates are women and 2 are men:

I W1, W2, W3, M1, M2

I Let X be a random variable that is the # of women hired I What is the PMF of X ?

I What is the CMF of X ?

Sex Discrimination Example

I There are 5 candidates for 2 job openings I 3 of the candidates are women and 2 are men:

I W1, W2, W3, M1, M2

I Let X be a random variable that is the # of women hired

I X ∈ {0, 1, 2} I What is the CMF of X ?

Sex Discrimination Example

I There are 5 candidates for 2 job openings I 3 of the candidates are women and 2 are men:

I W1, W2, W3, M1, M2

I Let X be a random variable that is the # of women hired

I X ∈ {0, 1, 2}

I What is the PMF of X ? Sex Discrimination Example

I There are 5 candidates for 2 job openings I 3 of the candidates are women and 2 are men:

I W1, W2, W3, M1, M2

I Let X be a random variable that is the # of women hired

I X ∈ {0, 1, 2}

I What is the PMF of X ?

I What is the CMF of X ? 10 possibilities in the sample space:

I (W1, W2) (W1, W3) (W1, M1) (W1, M2)

I (W2, W3) (W2, M1) (W2, M2)

I (W3, M1) (W3, M2)

I (M1, M2)

Sex Discrimination Example I (W1, W2) (W1, W3) (W1, M1) (W1, M2)

I (W2, W3) (W2, M1) (W2, M2)

I (W3, M1) (W3, M2)

I (M1, M2)

Sex Discrimination Example

10 possibilities in the sample space: I (W2, W3) (W2, M1) (W2, M2)

I (W3, M1) (W3, M2)

I (M1, M2)

Sex Discrimination Example

10 possibilities in the sample space:

I (W1, W2) (W1, W3) (W1, M1) (W1, M2) I (W3, M1) (W3, M2)

I (M1, M2)

Sex Discrimination Example

10 possibilities in the sample space:

I (W1, W2) (W1, W3) (W1, M1) (W1, M2)

I (W2, W3) (W2, M1) (W2, M2) I (M1, M2)

Sex Discrimination Example

10 possibilities in the sample space:

I (W1, W2) (W1, W3) (W1, M1) (W1, M2)

I (W2, W3) (W2, M1) (W2, M2)

I (W3, M1) (W3, M2) Sex Discrimination Example

10 possibilities in the sample space:

I (W1, W2) (W1, W3) (W1, M1) (W1, M2)

I (W2, W3) (W2, M1) (W2, M2)

I (W3, M1) (W3, M2)

I (M1, M2) Sex Discrimination Example

10 possibilities in the sample space:

I (W1, W2)(W1, W3)(W1, M1) (W1, M2)

I (W2, W3)(W2, M1) (W2, M2)

I (W3, M1) (W3, M2)

I (M1, M2) I (W1, W2) (W1, W3) (W1, M1) (W1, M2)

I (W2, W3) (W2, M1) (W2, M2)

I (W3, M1) (W3, M2)

I (M1, M2)

I Probability mass function distribution: X = 0 X = 1 X = 2 f (x) ???

I Cumulative mass function distribution: X 6 0 X 6 1 X 6 2 F (x) ???

Sex Discrimination Example I Probability mass function distribution: X = 0 X = 1 X = 2 f (x) ???

I Cumulative mass function distribution: X 6 0 X 6 1 X 6 2 F (x) ???

Sex Discrimination Example

I (W1, W2) (W1, W3) (W1, M1) (W1, M2)

I (W2, W3) (W2, M1) (W2, M2)

I (W3, M1) (W3, M2)

I (M1, M2) X = 0 X = 1 X = 2 f (x) ???

I Cumulative mass function distribution: X 6 0 X 6 1 X 6 2 F (x) ???

Sex Discrimination Example

I (W1, W2) (W1, W3) (W1, M1) (W1, M2)

I (W2, W3) (W2, M1) (W2, M2)

I (W3, M1) (W3, M2)

I (M1, M2)

I Probability mass function distribution: I Cumulative mass function distribution: X 6 0 X 6 1 X 6 2 F (x) ???

Sex Discrimination Example

I (W1, W2) (W1, W3) (W1, M1) (W1, M2)

I (W2, W3) (W2, M1) (W2, M2)

I (W3, M1) (W3, M2)

I (M1, M2)

I Probability mass function distribution: X = 0 X = 1 X = 2 f (x) ??? X 6 0 X 6 1 X 6 2 F (x) ???

Sex Discrimination Example

I (W1, W2) (W1, W3) (W1, M1) (W1, M2)

I (W2, W3) (W2, M1) (W2, M2)

I (W3, M1) (W3, M2)

I (M1, M2)

I Probability mass function distribution: X = 0 X = 1 X = 2 f (x) ???

I Cumulative mass function distribution: Sex Discrimination Example

I (W1, W2) (W1, W3) (W1, M1) (W1, M2)

I (W2, W3) (W2, M1) (W2, M2)

I (W3, M1) (W3, M2)

I (M1, M2)

I Probability mass function distribution: X = 0 X = 1 X = 2 f (x) ???

I Cumulative mass function distribution: X 6 0 X 6 1 X 6 2 F (x) ??? Sex Discrimination Example

I (W1, W2) (W1, W3) (W1, M1) (W1, M2)

I (W2, W3) (W2, M1) (W2, M2)

I (W3, M1) (W3, M2)

I (M1, M2)

I Probability mass function distribution: X = 0 X = 1 X = 2 f (x) ???

I Cumulative mass function distribution: X 6 0 X 6 1 X 6 2 F (x) ??? Sex Discrimination Example

I (W1, W2) (W1, W3) (W1, M1) (W1, M2)

I (W2, W3) (W2, M1) (W2, M2)

I (W3, M1) (W3, M2)

I (M1, M2)

I Probability mass function distribution: X = 0 X = 1 X = 2 f (x) 1/10 ? ?

I Cumulative mass function distribution: X 6 0 X 6 1 X 6 2 F (x) ??? Sex Discrimination Example

I (W1, W2) (W1, W3) (W1, M1)(W1, M2)

I (W2, W3) (W2, M1)(W2, M2)

I (W3, M1)(W3, M2)

I (M1, M2)

I Probability mass function distribution: X = 0 X = 1 X = 2 f (x) 1/10 ? ?

I Cumulative mass function distribution: X 6 0 X 6 1 X 6 2 F (x) ??? Sex Discrimination Example

I (W1, W2) (W1, W3) (W1, M1)(W1, M2)

I (W2, W3) (W2, M1)(W2, M2)

I (W3, M1)(W3, M2)

I (M1, M2)

I Probability mass function distribution: X = 0 X = 1 X = 2 f (x) 1/10 6/10 ?

I Cumulative mass function distribution: X 6 0 X 6 1 X 6 2 F (x) ??? Sex Discrimination Example

I (W1, W2)(W1, W3) (W1, M1) (W1, M2)

I (W2, W3) (W2, M1) (W2, M2)

I (W3, M1) (W3, M2)

I (M1, M2)

I Probability mass function distribution: X = 0 X = 1 X = 2 f (x) 1/10 6/10 ?

I Cumulative mass function distribution: X 6 0 X 6 1 X 6 2 F (x) ??? Sex Discrimination Example

I (W1, W2)(W1, W3) (W1, M1) (W1, M2)

I (W2, W3) (W2, M1) (W2, M2)

I (W3, M1) (W3, M2)

I (M1, M2)

I Probability mass function distribution: X = 0 X = 1 X = 2 f (x) 1/10 6/10 3/10

I Cumulative mass function distribution: X 6 0 X 6 1 X 6 2 F (x) ??? Sex Discrimination Example

I (W1, W2)(W1, W3) (W1, M1) (W1, M2)

I (W2, W3) (W2, M1) (W2, M2)

I (W3, M1) (W3, M2)

I (M1, M2)

I Probability mass function distribution: X = 0 X = 1 X = 2 f (x) 1/10 6/10 3/10

I Cumulative mass function distribution: X 6 0 X 6 1 X 6 2 F (x) 1/10 ? ? Sex Discrimination Example

I (W1, W2)(W1, W3) (W1, M1) (W1, M2)

I (W2, W3) (W2, M1) (W2, M2)

I (W3, M1) (W3, M2)

I (M1, M2)

I Probability mass function distribution: X = 0 X = 1 X = 2 f (x) 1/10 6/10 3/10

I Cumulative mass function distribution: X 6 0 X 6 1 X 6 2 F (x) 1/10 7/10 ? Sex Discrimination Example

I (W1, W2)(W1, W3) (W1, M1) (W1, M2)

I (W2, W3) (W2, M1) (W2, M2)

I (W3, M1) (W3, M2)

I (M1, M2)

I Probability mass function distribution: X = 0 X = 1 X = 2 f (x) 1/10 6/10 3/10

I Cumulative mass function distribution: X 6 0 X 6 1 X 6 2 F (x) 1/10 7/10 1 I Can use this information to address substantive questions

I Probability mass function distribution: X = 0 X = 1 X = 2 f (x) 1/10 6/10 3/10

I Cumulative mass function distribution: X 6 0 X 6 1 X 6 2 F (x) 1/10 7/10 1

I Would you be concerned if 1 or fewer women were hired?

I Would you be concerned if exactly 1 woman was hired?

I Would you be concerned if no women were hired?

Sex Discrimination Example I Probability mass function distribution: X = 0 X = 1 X = 2 f (x) 1/10 6/10 3/10

I Cumulative mass function distribution: X 6 0 X 6 1 X 6 2 F (x) 1/10 7/10 1

I Would you be concerned if 1 or fewer women were hired?

I Would you be concerned if exactly 1 woman was hired?

I Would you be concerned if no women were hired?

Sex Discrimination Example

I Can use this information to address substantive questions X = 0 X = 1 X = 2 f (x) 1/10 6/10 3/10

I Cumulative mass function distribution: X 6 0 X 6 1 X 6 2 F (x) 1/10 7/10 1

I Would you be concerned if 1 or fewer women were hired?

I Would you be concerned if exactly 1 woman was hired?

I Would you be concerned if no women were hired?

Sex Discrimination Example

I Can use this information to address substantive questions

I Probability mass function distribution: I Cumulative mass function distribution: X 6 0 X 6 1 X 6 2 F (x) 1/10 7/10 1

I Would you be concerned if 1 or fewer women were hired?

I Would you be concerned if exactly 1 woman was hired?

I Would you be concerned if no women were hired?

Sex Discrimination Example

I Can use this information to address substantive questions

I Probability mass function distribution: X = 0 X = 1 X = 2 f (x) 1/10 6/10 3/10 X 6 0 X 6 1 X 6 2 F (x) 1/10 7/10 1

I Would you be concerned if 1 or fewer women were hired?

I Would you be concerned if exactly 1 woman was hired?

I Would you be concerned if no women were hired?

Sex Discrimination Example

I Can use this information to address substantive questions

I Probability mass function distribution: X = 0 X = 1 X = 2 f (x) 1/10 6/10 3/10

I Cumulative mass function distribution: I Would you be concerned if 1 or fewer women were hired?

I Would you be concerned if exactly 1 woman was hired?

I Would you be concerned if no women were hired?

Sex Discrimination Example

I Can use this information to address substantive questions

I Probability mass function distribution: X = 0 X = 1 X = 2 f (x) 1/10 6/10 3/10

I Cumulative mass function distribution: X 6 0 X 6 1 X 6 2 F (x) 1/10 7/10 1 I Would you be concerned if exactly 1 woman was hired?

I Would you be concerned if no women were hired?

Sex Discrimination Example

I Can use this information to address substantive questions

I Probability mass function distribution: X = 0 X = 1 X = 2 f (x) 1/10 6/10 3/10

I Cumulative mass function distribution: X 6 0 X 6 1 X 6 2 F (x) 1/10 7/10 1

I Would you be concerned if 1 or fewer women were hired? I Would you be concerned if no women were hired?

Sex Discrimination Example

I Can use this information to address substantive questions

I Probability mass function distribution: X = 0 X = 1 X = 2 f (x) 1/10 6/10 3/10

I Cumulative mass function distribution: X 6 0 X 6 1 X 6 2 F (x) 1/10 7/10 1

I Would you be concerned if 1 or fewer women were hired?

I Would you be concerned if exactly 1 woman was hired? Sex Discrimination Example

I Can use this information to address substantive questions

I Probability mass function distribution: X = 0 X = 1 X = 2 f (x) 1/10 6/10 3/10

I Cumulative mass function distribution: X 6 0 X 6 1 X 6 2 F (x) 1/10 7/10 1

I Would you be concerned if 1 or fewer women were hired?

I Would you be concerned if exactly 1 woman was hired?

I Would you be concerned if no women were hired? I Similar to a sample of data, we can calculate useful info about probability distribution:

I Expected value I Variance I Standard deviation

I Use Greek letters to denote these population parameters

I Can’t actually measure these empirically, but can calculate them based on what we think would happen if we repeated the random process over and over again

I Conceptually distinct from (but completely analogous to) sample parameters from earlier

Key features of probability distributions I Expected value I Variance I Standard deviation

I Use Greek letters to denote these population parameters

I Can’t actually measure these empirically, but can calculate them based on what we think would happen if we repeated the random process over and over again

I Conceptually distinct from (but completely analogous to) sample parameters from earlier

Key features of probability distributions

I Similar to a sample of data, we can calculate useful info about probability distribution: I Variance I Standard deviation

I Use Greek letters to denote these population parameters

I Can’t actually measure these empirically, but can calculate them based on what we think would happen if we repeated the random process over and over again

I Conceptually distinct from (but completely analogous to) sample parameters from earlier

Key features of probability distributions

I Similar to a sample of data, we can calculate useful info about probability distribution:

I Expected value I Standard deviation

I Use Greek letters to denote these population parameters

I Can’t actually measure these empirically, but can calculate them based on what we think would happen if we repeated the random process over and over again

I Conceptually distinct from (but completely analogous to) sample parameters from earlier

Key features of probability distributions

I Similar to a sample of data, we can calculate useful info about probability distribution:

I Expected value I Variance I Use Greek letters to denote these population parameters

I Can’t actually measure these empirically, but can calculate them based on what we think would happen if we repeated the random process over and over again

I Conceptually distinct from (but completely analogous to) sample parameters from earlier

Key features of probability distributions

I Similar to a sample of data, we can calculate useful info about probability distribution:

I Expected value I Variance I Standard deviation I Can’t actually measure these empirically, but can calculate them based on what we think would happen if we repeated the random process over and over again

I Conceptually distinct from (but completely analogous to) sample parameters from earlier

Key features of probability distributions

I Similar to a sample of data, we can calculate useful info about probability distribution:

I Expected value I Variance I Standard deviation

I Use Greek letters to denote these population parameters I Conceptually distinct from (but completely analogous to) sample parameters from earlier

Key features of probability distributions

I Similar to a sample of data, we can calculate useful info about probability distribution:

I Expected value I Variance I Standard deviation

I Use Greek letters to denote these population parameters

I Can’t actually measure these empirically, but can calculate them based on what we think would happen if we repeated the random process over and over again Key features of probability distributions

I Similar to a sample of data, we can calculate useful info about probability distribution:

I Expected value I Variance I Standard deviation

I Use Greek letters to denote these population parameters

I Can’t actually measure these empirically, but can calculate them based on what we think would happen if we repeated the random process over and over again

I Conceptually distinct from (but completely analogous to) sample parameters from earlier I Expected value: a measure of the center of a probability distribution

I Denoted as µ

E[X ] = µX = xp(x) all x X

I Sum of all possible values, each weighted by its relative probability

I Substantively: If we repeated random process repeatedly and averaged values together, what would we expect to ﬁnd?

Expected value I Denoted as µ

E[X ] = µX = xp(x) all x X

I Sum of all possible values, each weighted by its relative probability

I Substantively: If we repeated random process repeatedly and averaged values together, what would we expect to ﬁnd?

Expected value

I Expected value: a measure of the center of a probability distribution E[X ] = µX = xp(x) all x X

I Sum of all possible values, each weighted by its relative probability

I Substantively: If we repeated random process repeatedly and averaged values together, what would we expect to ﬁnd?

Expected value

I Expected value: a measure of the center of a probability distribution

I Denoted as µ I Sum of all possible values, each weighted by its relative probability

I Substantively: If we repeated random process repeatedly and averaged values together, what would we expect to ﬁnd?

Expected value

I Expected value: a measure of the center of a probability distribution

I Denoted as µ

E[X ] = µX = xp(x) all x X I Substantively: If we repeated random process repeatedly and averaged values together, what would we expect to ﬁnd?

Expected value

I Expected value: a measure of the center of a probability distribution

I Denoted as µ

E[X ] = µX = xp(x) all x X

I Sum of all possible values, each weighted by its relative probability Expected value

I Expected value: a measure of the center of a probability distribution

I Denoted as µ

E[X ] = µX = xp(x) all x X

I Sum of all possible values, each weighted by its relative probability

I Substantively: If we repeated random process repeatedly and averaged values together, what would we expect to ﬁnd? I Variance: Measure of the spread of a probability distribution:

V [X ] = σ2 = (x − µ)2p(x) all x X

I Standard deviation: Square root of the variance

SD[X ] = σ √ = σ2

Variance and Standard Deviation V [X ] = σ2 = (x − µ)2p(x) all x X

I Standard deviation: Square root of the variance

SD[X ] = σ √ = σ2

Variance and Standard Deviation

I Variance: Measure of the spread of a probability distribution: I Standard deviation: Square root of the variance

SD[X ] = σ √ = σ2

Variance and Standard Deviation

I Variance: Measure of the spread of a probability distribution:

V [X ] = σ2 = (x − µ)2p(x) all x X SD[X ] = σ √ = σ2

Variance and Standard Deviation

I Variance: Measure of the spread of a probability distribution:

V [X ] = σ2 = (x − µ)2p(x) all x X

I Standard deviation: Square root of the variance Variance and Standard Deviation

I Variance: Measure of the spread of a probability distribution:

V [X ] = σ2 = (x − µ)2p(x) all x X

I Standard deviation: Square root of the variance

SD[X ] = σ √ = σ2 Probability mass function distribution from earlier: X = 0 X = 1 X = 2 f (x) 1/10 6/10 3/10

1 6 3 µ = × 0 + × 1 + × 2 10 10 10 12 = 10

122 1 122 6 122 3 σ2 = 0 − + 1 − + 2 − 10 10 10 10 10 10 144 24 192 360 9 = + + = = 1000 1000 1000 1000 25

3 σ = 5

Sex Discrimination Example X = 0 X = 1 X = 2 f (x) 1/10 6/10 3/10

1 6 3 µ = × 0 + × 1 + × 2 10 10 10 12 = 10

122 1 122 6 122 3 σ2 = 0 − + 1 − + 2 − 10 10 10 10 10 10 144 24 192 360 9 = + + = = 1000 1000 1000 1000 25

3 σ = 5

Sex Discrimination Example Probability mass function distribution from earlier: 1 6 3 µ = × 0 + × 1 + × 2 10 10 10 12 = 10

122 1 122 6 122 3 σ2 = 0 − + 1 − + 2 − 10 10 10 10 10 10 144 24 192 360 9 = + + = = 1000 1000 1000 1000 25

3 σ = 5

Sex Discrimination Example Probability mass function distribution from earlier: X = 0 X = 1 X = 2 f (x) 1/10 6/10 3/10 122 1 122 6 122 3 σ2 = 0 − + 1 − + 2 − 10 10 10 10 10 10 144 24 192 360 9 = + + = = 1000 1000 1000 1000 25

3 σ = 5

Sex Discrimination Example Probability mass function distribution from earlier: X = 0 X = 1 X = 2 f (x) 1/10 6/10 3/10

1 6 3 µ = × 0 + × 1 + × 2 10 10 10 12 = 10 3 σ = 5

Sex Discrimination Example Probability mass function distribution from earlier: X = 0 X = 1 X = 2 f (x) 1/10 6/10 3/10

1 6 3 µ = × 0 + × 1 + × 2 10 10 10 12 = 10

122 1 122 6 122 3 σ2 = 0 − + 1 − + 2 − 10 10 10 10 10 10 144 24 192 360 9 = + + = = 1000 1000 1000 1000 25 Sex Discrimination Example Probability mass function distribution from earlier: X = 0 X = 1 X = 2 f (x) 1/10 6/10 3/10

1 6 3 µ = × 0 + × 1 + × 2 10 10 10 12 = 10

122 1 122 6 122 3 σ2 = 0 − + 1 − + 2 − 10 10 10 10 10 10 144 24 192 360 9 = + + = = 1000 1000 1000 1000 25

3 σ = 5 I Have focused on distribution of one random variable, X

I A joint probability mass function describes behavior of 2 discrete random variables, X and Y , at same time

I Lists probabilities for all pairs of possible values of X and Y

I Can represent as relative frequency table, joint frequency histogram, or formula

Joint Probability Distributions I A joint probability mass function describes behavior of 2 discrete random variables, X and Y , at same time

I Lists probabilities for all pairs of possible values of X and Y

I Can represent as relative frequency table, joint frequency histogram, or formula

Joint Probability Distributions

I Have focused on distribution of one random variable, X I Lists probabilities for all pairs of possible values of X and Y

I Can represent as relative frequency table, joint frequency histogram, or formula

Joint Probability Distributions

I Have focused on distribution of one random variable, X

I A joint probability mass function describes behavior of 2 discrete random variables, X and Y , at same time I Can represent as relative frequency table, joint frequency histogram, or formula

Joint Probability Distributions

I Have focused on distribution of one random variable, X

I A joint probability mass function describes behavior of 2 discrete random variables, X and Y , at same time

I Lists probabilities for all pairs of possible values of X and Y Joint Probability Distributions

I Have focused on distribution of one random variable, X

I A joint probability mass function describes behavior of 2 discrete random variables, X and Y , at same time

I Lists probabilities for all pairs of possible values of X and Y

I Can represent as relative frequency table, joint frequency histogram, or formula I Written as P(X = x, Y = y)

I Can also have conditional probabilities: (X = x|Y = y)

I And can use joint probability P(X = x, Y = y) to back out marginal probability distribution, P(X = x) or P(Y = y)

I Ex) Suppose in Boston there is relationship between ﬂoods & snowfall in a given year

Joint Probability Distributions I Can also have conditional probabilities: (X = x|Y = y)

I And can use joint probability P(X = x, Y = y) to back out marginal probability distribution, P(X = x) or P(Y = y)

I Ex) Suppose in Boston there is relationship between ﬂoods & snowfall in a given year

Joint Probability Distributions

I Written as P(X = x, Y = y) I And can use joint probability P(X = x, Y = y) to back out marginal probability distribution, P(X = x) or P(Y = y)

I Ex) Suppose in Boston there is relationship between ﬂoods & snowfall in a given year

Joint Probability Distributions

I Written as P(X = x, Y = y)

I Can also have conditional probabilities: (X = x|Y = y) I Ex) Suppose in Boston there is relationship between ﬂoods & snowfall in a given year

Joint Probability Distributions

I Written as P(X = x, Y = y)

I Can also have conditional probabilities: (X = x|Y = y)

I And can use joint probability P(X = x, Y = y) to back out marginal probability distribution, P(X = x) or P(Y = y) Joint Probability Distributions

I Written as P(X = x, Y = y)

I Can also have conditional probabilities: (X = x|Y = y)

I And can use joint probability P(X = x, Y = y) to back out marginal probability distribution, P(X = x) or P(Y = y)

I Ex) Suppose in Boston there is relationship between ﬂoods & snowfall in a given year Association between ﬂoods and snowfalls

# of Snowstorms (Y) 0 1 2 3 4 Total 0 0.15 0.05 0 0 0 0.2 1 0.1 0.1 0.05 0 0 0.25 # of Floods (X) 2 0.05 0.05 0.1 0.05 0 0.25 3 0 0.05 0.05 0.07 0.03 0.2 4 0 0 0 0.03 0.07 0.1 Total 0.3 0.25 0.2 0.15 0.1 1.00 Association between ﬂoods and snowfalls

I What is probability of 2 ﬂoods and 3 snowstorms in a randomly chosen year? Association between ﬂoods and snowfalls

I What is the marginal probability mass function of Y ? Association between ﬂoods and snowfalls

I What is the conditional probability mass function of Y conditional on X = 2? Association between ﬂoods and snowfalls

# of Snowstorms (Y) 0 1 2 3 4 Total 0 0.15 0.05 0 0 0 0.2 1 0.1 0.1 0.05 0 0 0.25 # of Floods (X) 2 0.20 0.20 0.40 0.20 0 1.00 3 0 0.05 0.05 0.07 0.03 0.2 4 0 0 0 0.03 0.07 0.1 Total 0.3 0.25 0.2 0.15 0.1 1.00

I What is the conditional probability mass function of Y conditional on X = 2? I We can write down some PMFs and CMFs for easy examples → Real world much more complicated

I Fortunately, don’t have to calculate our own PMFs/CMFs for every random variable we are interested in

I Generations of statisticians have derived generalizable PMFs, CMFs for many frequently occurring processes

I This includes processes frequently seen in social sciences/policy I Also includes processes frequently seen in natural sciences (e.g., Normal distribution) I → Wikipedia a great resource

Probability Distribution Families I Fortunately, don’t have to calculate our own PMFs/CMFs for every random variable we are interested in

I Generations of statisticians have derived generalizable PMFs, CMFs for many frequently occurring processes

I This includes processes frequently seen in social sciences/policy I Also includes processes frequently seen in natural sciences (e.g., Normal distribution) I → Wikipedia a great resource

Probability Distribution Families

I We can write down some PMFs and CMFs for easy examples → Real world much more complicated I Generations of statisticians have derived generalizable PMFs, CMFs for many frequently occurring processes

I This includes processes frequently seen in social sciences/policy I Also includes processes frequently seen in natural sciences (e.g., Normal distribution) I → Wikipedia a great resource

Probability Distribution Families

I We can write down some PMFs and CMFs for easy examples → Real world much more complicated

I Fortunately, don’t have to calculate our own PMFs/CMFs for every random variable we are interested in I This includes processes frequently seen in social sciences/policy I Also includes processes frequently seen in natural sciences (e.g., Normal distribution) I → Wikipedia a great resource

Probability Distribution Families

I We can write down some PMFs and CMFs for easy examples → Real world much more complicated

I Fortunately, don’t have to calculate our own PMFs/CMFs for every random variable we are interested in

I Generations of statisticians have derived generalizable PMFs, CMFs for many frequently occurring processes I Also includes processes frequently seen in natural sciences (e.g., Normal distribution) I → Wikipedia a great resource

Probability Distribution Families

I We can write down some PMFs and CMFs for easy examples → Real world much more complicated

I Fortunately, don’t have to calculate our own PMFs/CMFs for every random variable we are interested in

I Generations of statisticians have derived generalizable PMFs, CMFs for many frequently occurring processes

I This includes processes frequently seen in social sciences/policy I → Wikipedia a great resource

Probability Distribution Families

I We can write down some PMFs and CMFs for easy examples → Real world much more complicated

I Fortunately, don’t have to calculate our own PMFs/CMFs for every random variable we are interested in

I Generations of statisticians have derived generalizable PMFs, CMFs for many frequently occurring processes

I This includes processes frequently seen in social sciences/policy I Also includes processes frequently seen in natural sciences (e.g., Normal distribution) Probability Distribution Families

I We can write down some PMFs and CMFs for easy examples → Real world much more complicated

I Fortunately, don’t have to calculate our own PMFs/CMFs for every random variable we are interested in

I Generations of statisticians have derived generalizable PMFs, CMFs for many frequently occurring processes

I This includes processes frequently seen in social sciences/policy I Also includes processes frequently seen in natural sciences (e.g., Normal distribution) I → Wikipedia a great resource I A single coin ﬂip well-known example of one of most basic probability distributions, a Bernoulli distribution/trial

I Need:

I A trial (e.g., coin ﬂip) that has a probability of success (p) and a probability of failure (1 − p)

I For Bernoulli processes, the PMF:

p for x = 1 f (x) = 1 − p for x = 0

I which can be rewritten as

f (x) = px (1 − p)1−x

Bernoulli Distribution I Need:

I A trial (e.g., coin ﬂip) that has a probability of success (p) and a probability of failure (1 − p)

I For Bernoulli processes, the PMF:

p for x = 1 f (x) = 1 − p for x = 0

I which can be rewritten as

f (x) = px (1 − p)1−x

Bernoulli Distribution

I A single coin ﬂip well-known example of one of most basic probability distributions, a Bernoulli distribution/trial I A trial (e.g., coin ﬂip) that has a probability of success (p) and a probability of failure (1 − p)

I For Bernoulli processes, the PMF:

p for x = 1 f (x) = 1 − p for x = 0

I which can be rewritten as

f (x) = px (1 − p)1−x

Bernoulli Distribution

I A single coin ﬂip well-known example of one of most basic probability distributions, a Bernoulli distribution/trial

I Need: I For Bernoulli processes, the PMF:

p for x = 1 f (x) = 1 − p for x = 0

I which can be rewritten as

f (x) = px (1 − p)1−x

Bernoulli Distribution

I A single coin ﬂip well-known example of one of most basic probability distributions, a Bernoulli distribution/trial

I Need:

I A trial (e.g., coin ﬂip) that has a probability of success (p) and a probability of failure (1 − p) I which can be rewritten as

f (x) = px (1 − p)1−x

Bernoulli Distribution

I A single coin ﬂip well-known example of one of most basic probability distributions, a Bernoulli distribution/trial

I Need:

I A trial (e.g., coin ﬂip) that has a probability of success (p) and a probability of failure (1 − p)

I For Bernoulli processes, the PMF:

p for x = 1 f (x) = 1 − p for x = 0

Bernoulli Distribution

I A single coin ﬂip well-known example of one of most basic probability distributions, a Bernoulli distribution/trial

I Need:

I A trial (e.g., coin ﬂip) that has a probability of success (p) and a probability of failure (1 − p)

I For Bernoulli processes, the PMF:

p for x = 1 f (x) = 1 − p for x = 0

I which can be rewritten as

f (x) = px (1 − p)1−x I A series of Bernoulli trial results in a binomial distribution

I Ex) Series of coin ﬂips I Need:

I Only two possible outcomes at each trial (success and failure) I Constant probability of success, p, for each trial I A ﬁxed number of trials n that are independent I X represents the number of success in n trials

Binomial Distribution I Ex) Series of coin ﬂips I Need:

Binomial Distribution

I A series of Bernoulli trial results in a binomial distribution I Need:

Binomial Distribution

I A series of Bernoulli trial results in a binomial distribution

I Ex) Series of coin ﬂips I Only two possible outcomes at each trial (success and failure) I Constant probability of success, p, for each trial I A ﬁxed number of trials n that are independent I X represents the number of success in n trials

Binomial Distribution

I A series of Bernoulli trial results in a binomial distribution

I Ex) Series of coin ﬂips I Need: I Constant probability of success, p, for each trial I A ﬁxed number of trials n that are independent I X represents the number of success in n trials

Binomial Distribution

I A series of Bernoulli trial results in a binomial distribution

I Ex) Series of coin ﬂips I Need:

I Only two possible outcomes at each trial (success and failure) I A ﬁxed number of trials n that are independent I X represents the number of success in n trials

Binomial Distribution

I A series of Bernoulli trial results in a binomial distribution

I Ex) Series of coin ﬂips I Need:

I Only two possible outcomes at each trial (success and failure) I Constant probability of success, p, for each trial I X represents the number of success in n trials

Binomial Distribution

I A series of Bernoulli trial results in a binomial distribution

I Ex) Series of coin ﬂips I Need:

I Only two possible outcomes at each trial (success and failure) I Constant probability of success, p, for each trial I A ﬁxed number of trials n that are independent Binomial Distribution

I A series of Bernoulli trial results in a binomial distribution

I Ex) Series of coin ﬂips I Need:

I Has a PMF of: n P(X = x) = px (1 − p)n−x x

n I where x is the binomial coeﬃcient: gives us the possible ways of getting x successes in n trials n n! I x = x!(n−x)! I and n! = n × (n − 1) × (n − 2) × ... × (1)

I and a CMF of: n P(X x) = px (1 − p)n−x 6 x X

Binomial Distribution I Has a PMF of: n P(X = x) = px (1 − p)n−x x

n I where x is the binomial coeﬃcient: gives us the possible ways of getting x successes in n trials n n! I x = x!(n−x)! I and n! = n × (n − 1) × (n − 2) × ... × (1)

I and a CMF of: n P(X x) = px (1 − p)n−x 6 x X

Binomial Distribution

I For a binomial process, denote X ∼ Binomial(n, p) n P(X = x) = px (1 − p)n−x x

n I where x is the binomial coeﬃcient: gives us the possible ways of getting x successes in n trials n n! I x = x!(n−x)! I and n! = n × (n − 1) × (n − 2) × ... × (1)

I and a CMF of: n P(X x) = px (1 − p)n−x 6 x X

Binomial Distribution

I For a binomial process, denote X ∼ Binomial(n, p)

I Has a PMF of: n I where x is the binomial coeﬃcient: gives us the possible ways of getting x successes in n trials n n! I x = x!(n−x)! I and n! = n × (n − 1) × (n − 2) × ... × (1)

I and a CMF of: n P(X x) = px (1 − p)n−x 6 x X

Binomial Distribution

I For a binomial process, denote X ∼ Binomial(n, p)

I Has a PMF of: n P(X = x) = px (1 − p)n−x x n n! I x = x!(n−x)! I and n! = n × (n − 1) × (n − 2) × ... × (1)

I and a CMF of: n P(X x) = px (1 − p)n−x 6 x X

Binomial Distribution

I For a binomial process, denote X ∼ Binomial(n, p)

I Has a PMF of: n P(X = x) = px (1 − p)n−x x

n I where x is the binomial coeﬃcient: gives us the possible ways of getting x successes in n trials I and n! = n × (n − 1) × (n − 2) × ... × (1)

I and a CMF of: n P(X x) = px (1 − p)n−x 6 x X

Binomial Distribution

I For a binomial process, denote X ∼ Binomial(n, p)

I Has a PMF of: n P(X = x) = px (1 − p)n−x x

n I where x is the binomial coeﬃcient: gives us the possible ways of getting x successes in n trials n n! I x = x!(n−x)! I and a CMF of: n P(X x) = px (1 − p)n−x 6 x X

Binomial Distribution

I For a binomial process, denote X ∼ Binomial(n, p)

I Has a PMF of: n P(X = x) = px (1 − p)n−x x

n I where x is the binomial coeﬃcient: gives us the possible ways of getting x successes in n trials n n! I x = x!(n−x)! I and n! = n × (n − 1) × (n − 2) × ... × (1) n P(X x) = px (1 − p)n−x 6 x X

Binomial Distribution

I For a binomial process, denote X ∼ Binomial(n, p)

I Has a PMF of: n P(X = x) = px (1 − p)n−x x

n I where x is the binomial coeﬃcient: gives us the possible ways of getting x successes in n trials n n! I x = x!(n−x)! I and n! = n × (n − 1) × (n − 2) × ... × (1)

I and a CMF of: Binomial Distribution

I For a binomial process, denote X ∼ Binomial(n, p)

I Has a PMF of: n P(X = x) = px (1 − p)n−x x

n I where x is the binomial coeﬃcient: gives us the possible ways of getting x successes in n trials n n! I x = x!(n−x)! I and n! = n × (n − 1) × (n − 2) × ... × (1)

I and a CMF of: n P(X x) = px (1 − p)n−x 6 x X Binomial Distribution I Poisson Distribution: X is # of events per unit of time

I Probability that 25 aviation incidents occur in a given year I Probability that 5 MBTA buses stop at JFK bus stop in an hr

I Geometric Distribution: X is the # of Bernoulli trials needed to get one “success”

I Probability that HKS needs to ask 5 former ambassadors to join faculty before one says “yes” I Probability that Senate will conﬁrm 10 of President’s judicial appointments before it rejects a single one

I Many preprogrammed in Stata or R

I Wikipedia: Detailed explanations

Other useful discrete families I Probability that 25 aviation incidents occur in a given year I Probability that 5 MBTA buses stop at JFK bus stop in an hr

I Geometric Distribution: X is the # of Bernoulli trials needed to get one “success”

I Many preprogrammed in Stata or R

I Wikipedia: Detailed explanations

Other useful discrete families

I Poisson Distribution: X is # of events per unit of time I Probability that 5 MBTA buses stop at JFK bus stop in an hr

I Geometric Distribution: X is the # of Bernoulli trials needed to get one “success”

I Many preprogrammed in Stata or R

I Wikipedia: Detailed explanations

Other useful discrete families

I Poisson Distribution: X is # of events per unit of time

I Probability that 25 aviation incidents occur in a given year I Geometric Distribution: X is the # of Bernoulli trials needed to get one “success”

I Many preprogrammed in Stata or R

I Wikipedia: Detailed explanations

Other useful discrete families

I Poisson Distribution: X is # of events per unit of time

I Probability that 25 aviation incidents occur in a given year I Probability that 5 MBTA buses stop at JFK bus stop in an hr I Probability that HKS needs to ask 5 former ambassadors to join faculty before one says “yes” I Probability that Senate will conﬁrm 10 of President’s judicial appointments before it rejects a single one

I Many preprogrammed in Stata or R

I Wikipedia: Detailed explanations

Other useful discrete families

I Poisson Distribution: X is # of events per unit of time

I Probability that 25 aviation incidents occur in a given year I Probability that 5 MBTA buses stop at JFK bus stop in an hr

I Geometric Distribution: X is the # of Bernoulli trials needed to get one “success” I Probability that HKS needs to ask 5 former ambassadors to join faculty before one says “yes” I Probability that Senate will conﬁrm 10 of President’s judicial appointments before it rejects a single one

I Many preprogrammed in Stata or R

I Wikipedia: Detailed explanations

Other useful discrete families

I Poisson Distribution: X is # of events per unit of time

I Probability that 25 aviation incidents occur in a given year I Probability that 5 MBTA buses stop at JFK bus stop in an hr

I Geometric Distribution: X is the # of Bernoulli trials needed to get one “success” I Probability that Senate will conﬁrm 10 of President’s judicial appointments before it rejects a single one

I Many preprogrammed in Stata or R

I Wikipedia: Detailed explanations

Other useful discrete families

I Poisson Distribution: X is # of events per unit of time

I Probability that 25 aviation incidents occur in a given year I Probability that 5 MBTA buses stop at JFK bus stop in an hr

I Geometric Distribution: X is the # of Bernoulli trials needed to get one “success”

I Probability that HKS needs to ask 5 former ambassadors to join faculty before one says “yes” I Many preprogrammed in Stata or R

I Wikipedia: Detailed explanations

Other useful discrete families

I Poisson Distribution: X is # of events per unit of time

I Probability that 25 aviation incidents occur in a given year I Probability that 5 MBTA buses stop at JFK bus stop in an hr

I Geometric Distribution: X is the # of Bernoulli trials needed to get one “success”

Other useful discrete families

I Poisson Distribution: X is # of events per unit of time

I Probability that 25 aviation incidents occur in a given year I Probability that 5 MBTA buses stop at JFK bus stop in an hr

I Geometric Distribution: X is the # of Bernoulli trials needed to get one “success”

I Many preprogrammed in Stata or R Other useful discrete families

I Poisson Distribution: X is # of events per unit of time

I Probability that 25 aviation incidents occur in a given year I Probability that 5 MBTA buses stop at JFK bus stop in an hr

I Geometric Distribution: X is the # of Bernoulli trials needed to get one “success”

I Many preprogrammed in Stata or R

I Wikipedia: Detailed explanations Next Time

I Continuous probability distributions