Uniform and Exponential Distributions Notes Lecture 19 Uniform and Exponential

Distributions Continuous Text: A Course in Probability by Weiss 8.4 Random Variable Uniform Distribution

STAT 225 Introduction to Probability Models Exponential March 24, 2014 Distribution

Whitney Huang Purdue University

19.1

Uniform and Agenda Exponential Distributions Notes

1 Continuous Random Variable Continuous Random Variable

Uniform Distribution 2 Uniform Distribution Exponential Distribution

3 Exponential Distribution

19.2 Uniform and Continuous Random Variable Exponential Distributions Notes

Probability Density Function (pdf) 0.25

Continuous

0.20 Random Variable

Uniform Distribution 0.15 f(x) Exponential Distribution 0.10 0.05 0.00

0 2 4 6 8 10

x

Let X be a continuous random variable P(X = x) = 0 for all possible x R b P(a ≤ X ≤ b) = a fX (x) dx = FX (b) − FX (a) where R x 19.3 FX (x) = P(X ≤ x) = −∞ fX (x) dx

Uniform and Uniform Distribution Exponential Distributions Notes Characteristics of the Uniform random variable: Let X be a Uniform r.v. The definition of X: A variable that is evenly distributed over an interval

The support (possible values for X): The interval Continuous Random Variable between a and b Uniform Its parameter(s) and definition(s): a and b are the Distribution Exponential lower and upper bound respectively of the distribution Distribution 1 The probability density function (pdf): b−a for a ≤ x ≤ b x−a The cumulative distribution function (cdf): b−a for a ≤ x ≤ b, 0 for x < a, 1 for x > b a+b The expected value: E[X] = 2 (b−a)2 The variance: Var(X) = 12

19.4 Uniform and Uniform Distribution cont’d Exponential Distributions Notes Let X ∼ Uniform[1, 5]

Probability Density Function (pdf)

0.30 | | | | | | | | | | Continuous | | f(x) 0.15 | | Random Variable | | | | | | | | Uniform

0.00 Distribution 0 1 2 3 4 5 6 Exponential Distribution x

Cumulative Distribution Function (cdf) 0.8 f(x) 0.4 0.0 0 1 2 3 4 5 6

x

19.5

Uniform and Example 48 Exponential Distributions Notes Let X denote a number selected at random from the interval (0, 10) 1 Name the distribution and the parameter(s) 2 Calculate the expected values and variances Continuous Random Variable 3 Compute the 41st percentile Uniform Distribution

Exponential Distribution

19.6 Uniform and Example 48 cont’d Exponential Distributions Notes Solution.

Continuous Random Variable

Uniform Distribution

Exponential Distribution

19.7

Uniform and Example 49 Exponential Distributions Notes

Continuous Random Variable

Uniform Distribution

Exponential Distribution

Shaggy feeds Scooby a Scooby–snack after every hi–jinks that Scooby foils. Suppose Scooby foils a hi–jinks anywhere from 0 minutes into the show up until 15 minutes into the show. Let X be the amount of time until

Scooby receives a Scooby–snack 19.8 Uniform and Example 49 cont’d Exponential Distributions Notes

1 Name the distribution and the parameter(s) 2 Find the pdf, cdf, expected value, and variance for X 3 Compute P(X < 5) Continuous P(X > 10) Random Variable P(3 < X < 11) Uniform P(X < 12|X > 4) Distribution Exponential Distribution

19.9

Uniform and Example 49 cont’d Exponential Distributions Notes Solution.

Continuous Random Variable

Uniform Distribution

Exponential Distribution

19.10 Uniform and Exponential Distribution Exponential Distributions Notes often used as the distribution for the time required to complete a certain task or for the elapsed time between successive occurrences of a specified event used to model the behavior of units that have a

constant failure rate Continuous Random Variable can be thought of as the continuous analog of the Uniform geometric random variable Distribution Exponential Distribution

19.11

Uniform and Exponential Distribution Exponential Distributions Notes Characteristics of the Exponential random variable: Let X be a Exponential r.v. The definition of X: The time up until the first (or next) success

The support (possible values for X): [0, ∞) Continuous Random Variable

Its parameter(s) and definition(s): λ : the average Uniform rate or how many successes per time unit Distribution −λ Exponential The probability density function (pdf): λe x for x ≥ 0 Distribution The cumulative distribution function (cdf):1 − e−λx for x ≥ 0 1 The expected value: E[X] = λ 1 The variance: Var(X) = λ2 Memoryless property: P(X > s + t|X > s) = P(X > t) Tail Probability: P(X > t) = e−λt

19.12 Uniform and Exponential Distribution cont’d Exponential Distributions Notes Let X ∼ Exp(1)

Probability Density Function (pdf) 0.8 Continuous

f(x) Random Variable 0.4

Uniform 0.0 Distribution 0 2 4 6 8 Exponential Distribution x

Cumulative Distribution Function (cdf) 0.8 f(x) 0.4 0.0 0 2 4 6 8

x

19.13

Uniform and Example 50 Exponential Distributions Notes Suppose the time it takes a puppy to run and get a ball, say T , follows an exponential distribution with a mean of 30 seconds. 1 State the distribution and parameters of T Continuous 2 What is the probability that it takes the puppy more Random Variable than 50 seconds to get the ball? Uniform Distribution 3 Assuming independence, what is the probability that Exponential it takes the puppy less than 40 seconds to fetch each Distribution of the next 5 balls? 4 What is the probability that it will take the puppy more than 45 seconds to get the ball knowing that it took the puppy longer than 20 seconds?

19.14 Uniform and Example 50 cont’d Exponential Distributions Notes Solution.

Continuous Random Variable

Uniform Distribution

Exponential Distribution

19.15

Notes