Random Variables STATISTICS – Lecture No

Random Variables STATISTICS – Lecture no. 3

Jiˇr´ıNeubauer

Department of Econometrics FEM UO Brno oﬃce 69a, tel. 973 442029 email:[email protected]

13. 10. 2009

Jiˇr´ıNeubauer Random Variables Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Random Variable

Many random experiments have numerical outcomes. Deﬁnition A random variable is a real-valued function X (ω) deﬁned on the sample space Ω.

The set of possible values of the random variable X is called the range of X . M = {x; X (ω) = x}.

Jiˇr´ıNeubauer Random Variables the number of dots when a die is rolled, the range is M = {1, 2,... 6} the number of rolls of a die until the ﬁrst 6 appears, the range is M = {1, 2, }˙ the lifetime of the lightbulb, the range is M = {x; x ≥ 0},

Examples of random variables:

Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Random Variable

Jiˇr´ıNeubauer Random Variables Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Random Variable

We denote random variables by capital letters X , Y ,... (eventually X1, X2,... ) and their particular values by small letters x, y,... . Using random variables we can describe random events, for example X = x, X ≤ x, x1 < X < x2 etc. Examples of random variables: the number of dots when a die is rolled, the range is M = {1, 2,... 6} the number of rolls of a die until the ﬁrst 6 appears, the range is M = {1, 2, }˙ the lifetime of the lightbulb, the range is M = {x; x ≥ 0},

Jiˇr´ıNeubauer Random Variables continuous ... M is a closed or open interval.

Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Random Variable

According to the range M we separate random variables to discrete ... M is ﬁnite or countable,

Jiˇr´ıNeubauer Random Variables Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Random Variable

According to the range M we separate random variables to discrete ... M is ﬁnite or countable, continuous ... M is a closed or open interval.

Jiˇr´ıNeubauer Random Variables Examples of continuous random variables: the height of a person, M = (0, ∞) the time taken to complete an examination, M = (0, ∞) the amount of milk in a bottle, M = (0, ∞)

Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Random Variable

Examples of discrete random variables: the number of cars sold at a dealership during a given month, M = {0, 1, 2,... } the number of houses in a certain block, M = {1, 2,... } the number of ﬁsh caught on a ﬁshing trip, M = {0, 1, 2,... } the number of heads obtained in three tosses of a coin, M = {0, 1, 2, 3}

Jiˇr´ıNeubauer Random Variables Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Random Variable

Examples of continuous random variables: the height of a person, M = (0, ∞) the time taken to complete an examination, M = (0, ∞) the amount of milk in a bottle, M = (0, ∞)

Jiˇr´ıNeubauer Random Variables and some measures: measures of location, measures of dispersion, measures of concentration.

Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Random Variable

For the description of random variables we will use some functions:

a (cumulative) distribution function F (x), a probability function p(x) – only for discrete random variables, a probability density function f (x) – only for continuous random variables.

Jiˇr´ıNeubauer Random Variables Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Random Variable

For the description of random variables we will use some functions:

a (cumulative) distribution function F (x), a probability function p(x) – only for discrete random variables, a probability density function f (x) – only for continuous random variables.

and some measures: measures of location, measures of dispersion, measures of concentration.

Jiˇr´ıNeubauer Random Variables Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Distribution Function

Deﬁnition Let X be any random variable. The distribution function F (x) of the random variable X is deﬁned as

F (x) = P(X ≤ x), x ∈ R.

Note: distribution function = cumulative distribution function

Jiˇr´ıNeubauer Random Variables F (x) is a non-decreasing, right-continuous function, it has limits

lim F (x) = 0, lim F (x) = 1, x→−∞ x→∞

if range of X is M = {x; x ∈ (a, bi} then F (a) = 0 a F (b) = 1,

for every real numbers x1 and x2: P(x1 < X ≤ x2) = F (x2) − F (x1).

Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Distribution Function

We mention some important properties of F (x): for every real x: 0 ≤ F (x) ≤ 1,

Jiˇr´ıNeubauer Random Variables it has limits

lim F (x) = 0, lim F (x) = 1, x→−∞ x→∞

if range of X is M = {x; x ∈ (a, bi} then F (a) = 0 a F (b) = 1,

for every real numbers x1 and x2: P(x1 < X ≤ x2) = F (x2) − F (x1).

Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Distribution Function

We mention some important properties of F (x): for every real x: 0 ≤ F (x) ≤ 1, F (x) is a non-decreasing, right-continuous function,

Jiˇr´ıNeubauer Random Variables for every real numbers x1 and x2: P(x1 < X ≤ x2) = F (x2) − F (x1).

Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Distribution Function

We mention some important properties of F (x): for every real x: 0 ≤ F (x) ≤ 1, F (x) is a non-decreasing, right-continuous function, it has limits

lim F (x) = 0, lim F (x) = 1, x→−∞ x→∞

if range of X is M = {x; x ∈ (a, bi} then F (a) = 0 a F (b) = 1,

Jiˇr´ıNeubauer Random Variables Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Distribution Function

We mention some important properties of F (x): for every real x: 0 ≤ F (x) ≤ 1, F (x) is a non-decreasing, right-continuous function, it has limits

lim F (x) = 0, lim F (x) = 1, x→−∞ x→∞

if range of X is M = {x; x ∈ (a, bi} then F (a) = 0 a F (b) = 1,

for every real numbers x1 and x2: P(x1 < X ≤ x2) = F (x2) − F (x1).

Jiˇr´ıNeubauer Random Variables Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Discrete Random Variable

For a discrete random variable X , we are interested in computing probabilities of the type P(X = xk ) for various values xk in range of X . Deﬁnition

Let X be a discrete random variable with range {x1, x2,... } (ﬁnite or countably inﬁnite). The function

p(x) = P(X = x)

is called the probability function of X .

Note: probability function = probability mass function

Jiˇr´ıNeubauer Random Variables X p(x) = 1 x∈M

for every two real numbers xk and xl (xk ≤ xl ):

x Xl P(xk ≤ X ≤ xl ) = p(xi ). xi =xk

Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Discrete Random Variable

We mention some important properties of p(x) for every real number x, 0 ≤ p(x) ≤ 1,

Jiˇr´ıNeubauer Random Variables for every two real numbers xk and xl (xk ≤ xl ):

x Xl P(xk ≤ X ≤ xl ) = p(xi ). xi =xk

Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Discrete Random Variable

We mention some important properties of p(x) for every real number x, 0 ≤ p(x) ≤ 1,

X p(x) = 1 x∈M

Jiˇr´ıNeubauer Random Variables Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Discrete Random Variable

We mention some important properties of p(x) for every real number x, 0 ≤ p(x) ≤ 1,

X p(x) = 1 x∈M

for every two real numbers xk and xl (xk ≤ xl ):

x Xl P(xk ≤ X ≤ xl ) = p(xi ). xi =xk

Jiˇr´ıNeubauer Random Variables Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Probability Function

The probability function p(x) can be described by the table, P X x1 x2 ... xi ... p(x) p(x1) p(x2) ... p(xi ) ... 1

Jiˇr´ıNeubauer Random Variables Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Probability Function

the graph [x, p(x)],

Jiˇr´ıNeubauer Random Variables Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Probability Function

the formula, for example

π(1 − π)x x = 0, 1, 2,..., p(x) = 0 otherwise,

where π is a given probability.

Jiˇr´ıNeubauer Random Variables Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example

The shooter has 3 bullets and shoots at the target until the first hit or until the last bullet. The probability that the shooter hits the target after one shot is 0.6. The random variable X is the number of the fired bullets. Find the probability and the distribution function of the given random variable. What is the probability that the number of the fired bullets will not be larger then 2?

Jiˇr´ıNeubauer Random Variables p(2) = P(X = 2) = 0.4 · 0.6 = 0.24, p(3) = P(X = 3) = 0.4·0.4·0.6+0.4·0.4·0.4 = 0.4·0.4 = 0.16.

Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example

Random variable X is discrete with the range M = {1, 2, 3}. The probability function is: p(1) = P(X = 1) = 0.6,

Jiˇr´ıNeubauer Random Variables p(3) = P(X = 3) = 0.4·0.4·0.6+0.4·0.4·0.4 = 0.4·0.4 = 0.16.

Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example

Random variable X is discrete with the range M = {1, 2, 3}. The probability function is: p(1) = P(X = 1) = 0.6, p(2) = P(X = 2) = 0.4 · 0.6 = 0.24,

Jiˇr´ıNeubauer Random Variables Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example

Random variable X is discrete with the range M = {1, 2, 3}. The probability function is: p(1) = P(X = 1) = 0.6, p(2) = P(X = 2) = 0.4 · 0.6 = 0.24, p(3) = P(X = 3) = 0.4·0.4·0.6+0.4·0.4·0.4 = 0.4·0.4 = 0.16.

Jiˇr´ıNeubauer Random Variables Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example

All results are summarized in the table

x 1 2 3 P p(x) 0.6 0.24 0.16 1

The probability function can be described by the formula

 x−1  0.6 · 0.4 x = 1, 2, p(x) = 0.42 x = 3,  0 otherwise.

Jiˇr´ıNeubauer Random Variables F (1) = P(X ≤ 1) = p(1) = 0.6, F (1.5) = P(X ≤ 1.5) = P(X ≤ 1) = p(1) = 0.6, F (2) = P(X ≤ 2) = p(1) + p(2) = 0.84, F (3) = P(X ≤ 3) = p(1) + p(2) + p(3) = 1, F (4) = P(X ≤ 4) = p(1) + p(2) + p(3) = 1.

Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example

We can calculate some values of the distribution function F (x): F (0) = P(X ≤ 0) = 0,

Jiˇr´ıNeubauer Random Variables F (1.5) = P(X ≤ 1.5) = P(X ≤ 1) = p(1) = 0.6, F (2) = P(X ≤ 2) = p(1) + p(2) = 0.84, F (3) = P(X ≤ 3) = p(1) + p(2) + p(3) = 1, F (4) = P(X ≤ 4) = p(1) + p(2) + p(3) = 1.

Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example

We can calculate some values of the distribution function F (x): F (0) = P(X ≤ 0) = 0, F (1) = P(X ≤ 1) = p(1) = 0.6,

Jiˇr´ıNeubauer Random Variables F (2) = P(X ≤ 2) = p(1) + p(2) = 0.84, F (3) = P(X ≤ 3) = p(1) + p(2) + p(3) = 1, F (4) = P(X ≤ 4) = p(1) + p(2) + p(3) = 1.

Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example

We can calculate some values of the distribution function F (x): F (0) = P(X ≤ 0) = 0, F (1) = P(X ≤ 1) = p(1) = 0.6, F (1.5) = P(X ≤ 1.5) = P(X ≤ 1) = p(1) = 0.6,

Jiˇr´ıNeubauer Random Variables F (3) = P(X ≤ 3) = p(1) + p(2) + p(3) = 1, F (4) = P(X ≤ 4) = p(1) + p(2) + p(3) = 1.

Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example

We can calculate some values of the distribution function F (x): F (0) = P(X ≤ 0) = 0, F (1) = P(X ≤ 1) = p(1) = 0.6, F (1.5) = P(X ≤ 1.5) = P(X ≤ 1) = p(1) = 0.6, F (2) = P(X ≤ 2) = p(1) + p(2) = 0.84,

Jiˇr´ıNeubauer Random Variables F (4) = P(X ≤ 4) = p(1) + p(2) + p(3) = 1.

Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example

We can calculate some values of the distribution function F (x): F (0) = P(X ≤ 0) = 0, F (1) = P(X ≤ 1) = p(1) = 0.6, F (1.5) = P(X ≤ 1.5) = P(X ≤ 1) = p(1) = 0.6, F (2) = P(X ≤ 2) = p(1) + p(2) = 0.84, F (3) = P(X ≤ 3) = p(1) + p(2) + p(3) = 1,

Jiˇr´ıNeubauer Random Variables Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example

We can write  0 x < 1,   0.6 1 ≤ x < 2, F (x) = 0.84 2 ≤ x < 3,   1 x ≥ 3.

Jiˇr´ıNeubauer Random Variables Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example

Figure: The probability and the distribution function

Jiˇr´ıNeubauer Random Variables Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example

What is the probability that the number of the ﬁred bullets will not be larger then 2?

P(X ≤ 2) = P(X = 1) + P(X = 2) = p(1) + p(2) = F (2) = = 0.6 + 0.24 = 0.84.

Jiˇr´ıNeubauer Random Variables Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Probability Density Function

If the cumulative distribution function is a continuous function, then X is said to be a continuous random variable. Deﬁnition The probability density function of the random variable X is a non-negative function f (x) such that

x Z F (x) = f (t)dt, x ∈ R. −∞

Jiˇr´ıNeubauer Random Variables dF (x) 0 f (x) = dx = F (x), where the derivative exists, P(x1 ≤ X ≤ x2) = P(x1 < X < x2) = P(x1 < X ≤ x2) = x R2 P(x1 ≤ X < x2) = F (x2) − F (x1) = f (x)dx x1 If X is a continuous random variable, then P(X = x) = 0.

Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Probability Density Function

Some properties of f (x): ∞ R f (x)dx = R f (x)dx = 1, −∞ M

Jiˇr´ıNeubauer Random Variables P(x1 ≤ X ≤ x2) = P(x1 < X < x2) = P(x1 < X ≤ x2) = x R2 P(x1 ≤ X < x2) = F (x2) − F (x1) = f (x)dx x1 If X is a continuous random variable, then P(X = x) = 0.

Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Probability Density Function

Some properties of f (x): ∞ R f (x)dx = R f (x)dx = 1, −∞ M dF (x) 0 f (x) = dx = F (x), where the derivative exists,

Jiˇr´ıNeubauer Random Variables If X is a continuous random variable, then P(X = x) = 0.

Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Probability Density Function

Some properties of f (x): ∞ R f (x)dx = R f (x)dx = 1, −∞ M dF (x) 0 f (x) = dx = F (x), where the derivative exists, P(x1 ≤ X ≤ x2) = P(x1 < X < x2) = P(x1 < X ≤ x2) = x R2 P(x1 ≤ X < x2) = F (x2) − F (x1) = f (x)dx x1

The function f (x) we can describe by a formula or a graph, for example ( 1 − x−2 e 5 pro x > 2, f (x) = 5 0 pro x ≤ 2.

Jiˇr´ıNeubauer Random Variables Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example

The random variable X has the probability density function

cx2(1 − x) 0 < x < 1, f (x) = 0 otherwise.

Determine a constant c in order that f (x) is a probability density function. Find a distribution function of the random variable X . Calculate the probability P(0.2 < X < 0.8).

Jiˇr´ıNeubauer Random Variables Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example

The probability density function has to fulﬁl Z f (x)dx = 1.

1 1 1 R 2 R 2 3 h x3 x4 i cx (1 − x)dx = c (x − x )dx = c 3 − 4 = 0 0 0 1 1 c = c 3 − 4 = 12 = 1, we get c = 12.

Jiˇr´ıNeubauer Random Variables Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example

The distribution function can be calculated by the deﬁnition of the probability density function. We can write for 0 < x < 1

x x x R 2 R 2 3 h t3 t4 i F (x) = 12t (1 − t)dt = 12 (t − t )dt = 12 3 − 4 = 0 0 0 h x3 x4 i 3 4 = 12 3 − 4 = 4x − 3x .

  0 x ≤ 0, F (x) = x3(4 − 3x) 0 < x < 1,  1 otherwise.

Jiˇr´ıNeubauer Random Variables Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration

Figure: The probability density function and the distribution function

Jiˇr´ıNeubauer Random Variables If the distribution function is known, we can do simpler calculation

P(0.2 < X < 0.8) = F (0.8) − F (0.2) = = 0.83(4 − 3 · 0.8) − 0.23(4 − 3 · 0.2) = 0.792.

Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example

Using the probability density function we can calculate

0.8 Z 2 3 40.8 P(0.2 < X < 0.8) = 12x (1 − x)dx = 4x − 3x 0.2 = 0.792. 0.2

Jiˇr´ıNeubauer Random Variables Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example

Using the probability density function we can calculate

0.8 Z 2 3 40.8 P(0.2 < X < 0.8) = 12x (1 − x)dx = 4x − 3x 0.2 = 0.792. 0.2 If the distribution function is known, we can do simpler calculation

P(0.2 < X < 0.8) = F (0.8) − F (0.2) = = 0.83(4 − 3 · 0.8) − 0.23(4 − 3 · 0.2) = 0.792.

Jiˇr´ıNeubauer Random Variables Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example

A random variable X is described by the distribution function

0 x ≤ 0, F (x) = 1 − e−x x > 0.

Find a probability density function.

Jiˇr´ıNeubauer Random Variables Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example

Using mentioned formula

dF (x) f (x) = dx

d −x −x and the fact that dx (1 − e ) = e we get

0 x ≤ 0, f (x) = e−x x > 0.

Jiˇr´ıNeubauer Random Variables Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Measures of Location

The distribution function (the probability function or the probability density function) gives us the complete information about the random variable. Sometimes is useful to know some simpler and concentrated formulation of this information such as measures of location, dispersion and concentration. The best known measures of location are a mean (an expected value), quantiles (a median, an upper and a lower quartile, . . . ) and a mode.

Jiˇr´ıNeubauer Random Variables Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Expected Value

Deﬁnition The mean (the expected value) E(X ) of the random variable X (sometimes denoted as µ) is the value that is expected to occur per repetition, if an experiment is repeated a large number of times. For the discrete random variable is deﬁned as X E(X ) = xi p(xi ), M for the continuous random variable as Z E(X ) = xf (x)dx

M if the given sequence or integral absolutely converges. Jiˇr´ıNeubauer Random Variables the mean of the product of the constant c and the random variable X is equal to the product of the given constant c and the mean of X E(cX ) = cE(X ),

Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Expected Value

We mention some properties of the mean: the mean of the constant c is equal to this constant

E(c) = c,

Jiˇr´ıNeubauer Random Variables Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Expected Value

We mention some properties of the mean: the mean of the constant c is equal to this constant

E(c) = c,

the mean of the product of the constant c and the random variable X is equal to the product of the given constant c and the mean of X E(cX ) = cE(X ),

Jiˇr´ıNeubauer Random Variables if X1, X2,..., Xn are independent, then the mean of their product is equal to the product of their means

E(X1X2 ... Xn) = E(X1)E(X2) ... E(Xn).

Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Expected Value

the mean of the sum of random variables X1, X2,..., Xn is equal to the sum of the mean of the given random variables,

E(X1 + X2 + ··· + Xn) = E(X1) + E(X2) + ··· + E(Xn),

Jiˇr´ıNeubauer Random Variables Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Expected Value

the mean of the sum of random variables X1, X2,..., Xn is equal to the sum of the mean of the given random variables,

E(X1 + X2 + ··· + Xn) = E(X1) + E(X2) + ··· + E(Xn),

if X1, X2,..., Xn are independent, then the mean of their product is equal to the product of their means

E(X1X2 ... Xn) = E(X1)E(X2) ... E(Xn).

The random variables X1, X2,..., Xn are independent if and only if for any numbers x1, x2,..., xn ∈ R is

P(X1 ≤ x1, X2 ≤ x2,..., Xn ≤ xn) =

= P(X1 ≤ x1) · P(X2 ≤ x2) ··· P(Xn ≤ xn).

Let us have the random vector X = (X1, X2,..., Xn), whose components X1, X2,..., Xn are the random variables. F (x) = F (x1, x2,..., xn) = P(X1 ≤ x1, X2 ≤ x2,..., Xn ≤ xn) is the distribution function of the vector X and F (x1), F (x2),..., F (xn) are the distribution functions of the random variables X1, X2,..., Xn. The random variables X1, X2,..., Xn are independent if and only if

F (x1, x2,..., xn) = F (x1) · F (x2) ··· F (xn).

If X is the random vector whose components are the discrete random variables, the function p(x) = p(x1, x2,..., xn) = P(X1 = x1, X2 = x2,..., Xn = xn) is the probability function of the vector X, p(x1), p(x2),..., p(xn) are the probability functions of X1, X2,..., Xn, then: X1, X2,..., Xn are independent if and only if

p(x1, x2,..., xn) = p(x1) · p(x2) ··· p(xn). If X is the random vector whose components are the continuous random variables, the function f (x) = f (x1, x2,..., xn) is the probability density function of the vector X, f (x1), f (x2),..., f (xn) are the probability density functions of X1, X2,..., Xn, then: X1, X2,..., Xn are independent if and only if

f (x1, x2,..., xn) = f (x1) · f (x2) ··· f (xn).

Jiˇr´ıNeubauer Random Variables Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Quantile

Deﬁnition

100P% quantile xP of the random variable with the increasing distribution function is such value of the random variable that

P(X ≤ xP ) = F (xP ) = P, 0 < P < 1.

Jiˇr´ıNeubauer Random Variables Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Quantile

The quantile x0.50 we call median Me(X ), it fulﬁls P(X ≤ Me(X )) = P(X ≥ Me(X )) = 0.50. The quantile x0.25 is called the lower quartile, the quantile x0.75 is called the upper quartile. The selected quantiles of some important distributions are tabulated. Jiˇr´ıNeubauer Random Variables Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Mode

Deﬁnition The mode Mo(X ) is the value of the random variable with the highest probability (for the discrete random variable), or the value, where the function f (x) has the maximum (for the continuous random variable ).

Jiˇr´ıNeubauer Random Variables Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example

Find the mean (the expected value) and the mode of the random variable deﬁned as the number of ﬁred bullets (see the example before). The probability function is

 x−1  0.6 · 0.4 x = 1, 2, p(x) = 0.42 x = 3,  0 otherwise.

Jiˇr´ıNeubauer Random Variables The mode is the value of the given random variable with the highest probability which is Mo(X ) = 1, because p(1) = 0.6.

Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example

The mean (the expected value) we get using the formula from the deﬁnition of E(X )

3 X 0 1 2 E(x) = xi p(xi ) = 1 · 0.6 · 0.4 + 2 · 0.6 · 0.4 + 3 · 0.4 = 1.56. i=1

Jiˇr´ıNeubauer Random Variables Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example

The mean (the expected value) we get using the formula from the deﬁnition of E(X )

3 X 0 1 2 E(x) = xi p(xi ) = 1 · 0.6 · 0.4 + 2 · 0.6 · 0.4 + 3 · 0.4 = 1.56. i=1 The mode is the value of the given random variable with the highest probability which is Mo(X ) = 1, because p(1) = 0.6.

Jiˇr´ıNeubauer Random Variables Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example

The random variable X is described by the probability density function 12x2(1 − x) 0 < x < 1, f (x) = 0 otherwise. Find the mean (the expected value) and the mode.

Jiˇr´ıNeubauer Random Variables The mode is the maximum of the probability density function. We have to ﬁnd the maximum of f (x) on the interval 0 < x < 1, d 2 2 dx 12x (1 − x) = 12(2x − 3x ) = 0, x(2 − 3x) = 0, we get x = 0 or x = 2/3. The maximum of f (x) is in x = 2/3 ⇒ Mo(X ) = 2/3.

Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example

We calculate the mean using the deﬁnition formula

1 1 Z Z x4 x5 1 3 E(X ) = xf (x)dx = x·12x2(1−x) = 12 − = = 0.6. 4 5 0 5 0 0

Jiˇr´ıNeubauer Random Variables Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example

We calculate the mean using the deﬁnition formula

1 1 Z Z x4 x5 1 3 E(X ) = xf (x)dx = x·12x2(1−x) = 12 − = = 0.6. 4 5 0 5 0 0 The mode is the maximum of the probability density function. We have to ﬁnd the maximum of f (x) on the interval 0 < x < 1, d 2 2 dx 12x (1 − x) = 12(2x − 3x ) = 0, x(2 − 3x) = 0, we get x = 0 or x = 2/3. The maximum of f (x) is in x = 2/3 ⇒ Mo(X ) = 2/3.

Jiˇr´ıNeubauer Random Variables Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example

Find the median,the upper and the lower quartile of the random variable X with the distribution function 1 − 1 x > 1, F (x) = x3 0 x ≤ 1.

Jiˇr´ıNeubauer Random Variables median x = √ 1 = 1.260, 0.50 3 1−0.50 lower quartile x = √ 1 = 1.101, 0.25 3 1−0.25 upper quartile x = √ 1 = 1.587. 0.75 3 1−0.75

Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example

The quantile is deﬁned by the formula F (xP ) = P. 1 1 − 3 = P, xP

1 xP = √ . 3 1 − P

Jiˇr´ıNeubauer Random Variables Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example

The quantile is deﬁned by the formula F (xP ) = P. 1 1 − 3 = P, xP

1 xP = √ . 3 1 − P

median x = √ 1 = 1.260, 0.50 3 1−0.50 lower quartile x = √ 1 = 1.101, 0.25 3 1−0.25 upper quartile x = √ 1 = 1.587. 0.75 3 1−0.75

Jiˇr´ıNeubauer Random Variables Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Measures of Dispersion

The elementary and widely-used measures of dispersion are the variance and the standard deviation.

Jiˇr´ıNeubauer Random Variables Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Variance

Deﬁnition The variance D(X ) of the random variable X (sometimes denoted as σ2) is deﬁned by the formula

D(X ) = E [X − E(X )]2 .

The variance of the discrete random variable is given by

X 2 D(X ) = [xi − E(X )] p(xi ), M the variance of the continuous random variable is Z D(X ) = [x − E(X )]2f (x)dx.

M Jiˇr´ıNeubauer Random Variables D(kX ) = k2D(X ), D(X + Y ) = D(X ) + D(Y ), if X and Y are independent, D(X ) ≥ 0 for every random variable,

Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Variance

Some properties of the variance: D(k) = 0, where k is a constant,

Jiˇr´ıNeubauer Random Variables D(X + Y ) = D(X ) + D(Y ), if X and Y are independent, D(X ) ≥ 0 for every random variable,

Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Variance

Some properties of the variance: D(k) = 0, where k is a constant, D(kX ) = k2D(X ),

Jiˇr´ıNeubauer Random Variables D(X ) ≥ 0 for every random variable,

Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Variance

Some properties of the variance: D(k) = 0, where k is a constant, D(kX ) = k2D(X ), D(X + Y ) = D(X ) + D(Y ), if X and Y are independent,

Jiˇr´ıNeubauer Random Variables Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Variance

Some properties of the variance: D(k) = 0, where k is a constant, D(kX ) = k2D(X ), D(X + Y ) = D(X ) + D(Y ), if X and Y are independent, D(X ) ≥ 0 for every random variable,

Jiˇr´ıNeubauer Random Variables Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Variance

D(X ) = E(X 2) − E(X )2, D(X ) = E[X − E(X )]2 = E[X 2 − 2XE(X ) + E(X )2] = = E(X 2) − E[2XE(X )] + E[E(X )2] = = E(X 2) − 2E(X )E(X ) + E(X )2 = E(X 2) − E(X )2 For the discrete random variable

X 2 2 D(X ) = xi p(xi ) − E(X ) , M for the continuous random variable Z D(X ) = x2f (x)dx − E(X )2.

Jiˇr´ıNeubauer Random Variables Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Standard Deviation

Deﬁnition The standard deviation σ(X ) of the random variable X is deﬁned as the square root of the variance

σ(X ) = pD(X ).

The standard deviation has the same unit as the random variable X .

Jiˇr´ıNeubauer Random Variables Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration

Figure: Relation between the mean and the standard deviation

Jiˇr´ıNeubauer Random Variables Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example

Find the variance and the standard deviation of the random variable deﬁned as the number of ﬁred bullets (see the example before).

Jiˇr´ıNeubauer Random Variables 3 2 X 2 X 2 2 2 2 E(X ) = xi p(xi ) = xi p(xi ) = 1 ·0.6+2 ·0.24+3 ·0.16 = 3, M i=1 then D(X ) = 3 − 1.562 = 0.566. The standard deviation is the square root of the variance

σ(X ) = pD(X ) = 0.753.

Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example

The mean is E(X ) = 1.56 (see the previous example). For the purpose of calculation of the variance we use the formula

D(X ) = E(X 2) − E(X )2

Jiˇr´ıNeubauer Random Variables The standard deviation is the square root of the variance

σ(X ) = pD(X ) = 0.753.

Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example

The mean is E(X ) = 1.56 (see the previous example). For the purpose of calculation of the variance we use the formula

D(X ) = E(X 2) − E(X )2

3 2 X 2 X 2 2 2 2 E(X ) = xi p(xi ) = xi p(xi ) = 1 ·0.6+2 ·0.24+3 ·0.16 = 3, M i=1 then D(X ) = 3 − 1.562 = 0.566.

Jiˇr´ıNeubauer Random Variables Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example

The mean is E(X ) = 1.56 (see the previous example). For the purpose of calculation of the variance we use the formula

D(X ) = E(X 2) − E(X )2

3 2 X 2 X 2 2 2 2 E(X ) = xi p(xi ) = xi p(xi ) = 1 ·0.6+2 ·0.24+3 ·0.16 = 3, M i=1 then D(X ) = 3 − 1.562 = 0.566. The standard deviation is the square root of the variance

σ(X ) = pD(X ) = 0.753.

Jiˇr´ıNeubauer Random Variables Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example

Find the variance and the standard deviation of the random variable X with the probability density function

12x2(1 − x) 0 < x < 1, f (x) = 0 otherwise.

Jiˇr´ıNeubauer Random Variables 2 32 1 D(X ) = − = = 0.04. 5 5 25 The standard deviation is 1 σ(X ) = pD(X ) = = 0.2. 5

Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example

The mean is E(X ) = 3/5 (see the previous example).

D(X ) = E(X 2) − E(X )2

1 1 2 R 2 R 2 2 h x5 x6 i E(X ) = x f (x)dx = x · 12x (1 − x)dx = 12 5 − 6 M 0 0 2 = 5 = 0.4,

Jiˇr´ıNeubauer Random Variables Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example

The mean is E(X ) = 3/5 (see the previous example).

D(X ) = E(X 2) − E(X )2

1 1 2 R 2 R 2 2 h x5 x6 i E(X ) = x f (x)dx = x · 12x (1 − x)dx = 12 5 − 6 M 0 0 2 = 5 = 0.4, 2 32 1 D(X ) = − = = 0.04. 5 5 25 The standard deviation is 1 σ(X ) = pD(X ) = = 0.2. 5

Jiˇr´ıNeubauer Random Variables Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Measures of Concentration

We will focus on the measures describing the shape of random variables distribution (skewness and kurtosis). These measures are deﬁned by moments.

Jiˇr´ıNeubauer Random Variables Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Measures of Concentration

Deﬁnition th 0 The r moment µr of the random variable X is deﬁned by the formula 0 r µr (X ) = E(X ) for r = 1, 2,.... The r th moment of the discrete random variable is given by

0 X r µr (X ) = xi p(xi ), M

r th moment of the continuous random variable is Z 0 r µr (X ) = x f (x)dx. M

Jiˇr´ıNeubauer Random Variables Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Measures of Concentration

Deﬁnition th The r central moment µr of the random variable X is deﬁned by the formula

r µr (X ) = E[X − E(X )] for r = 2, 3,....

The r th central moment of the discrete random variable is given by

X r µr (X ) = [xi − E(X )] p(xi ), M

the r th central moment of the continuous random variable is Z r µr (X ) = [x − E(X )] f (x)dx. M Jiˇr´ıNeubauer Random Variables According to the values of the skewness we can tell whether distribution is symmetric or asymmetric.

α3 = 0, distribution is symmetric,

α3 < 0, distribution is skewed to the right,

α3 > 0, distribution is skewed to the left.

Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Skewness

Deﬁnition

The skewness α3(X ) is deﬁned by the formula µ (X ) α (X ) = 3 . 3 σ(X )3

Jiˇr´ıNeubauer Random Variables Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Skewness

Deﬁnition

The skewness α3(X ) is deﬁned by the formula µ (X ) α (X ) = 3 . 3 σ(X )3

According to the values of the skewness we can tell whether distribution is symmetric or asymmetric.

α3 = 0, distribution is symmetric,

α3 < 0, distribution is skewed to the right,

α3 > 0, distribution is skewed to the left.

Jiˇr´ıNeubauer Random Variables Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Skewness

Figure: The skewness Jiˇr´ıNeubauer Random Variables Kurtosis is a measure of how outlier-prone a distribution is. The kurtosis of the normal distribution is 0. Distributions that are more outlier-prone than the normal distribution have kurtosis greater than 0; distributions that are less outlier-prone have kurtosis less than 0.

Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Kurtosis

Deﬁnition

The kurtosis α4(X ) is deﬁned by the formula µ (X ) α (X ) = 4 − 3. 4 σ(X )4

Jiˇr´ıNeubauer Random Variables Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Kurtosis

Deﬁnition

The kurtosis α4(X ) is deﬁned by the formula µ (X ) α (X ) = 4 − 3. 4 σ(X )4

Kurtosis is a measure of how outlier-prone a distribution is. The kurtosis of the normal distribution is 0. Distributions that are more outlier-prone than the normal distribution have kurtosis greater than 0; distributions that are less outlier-prone have kurtosis less than 0.

Jiˇr´ıNeubauer Random Variables Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Kurtosis

Figure: The kurtosis Jiˇr´ıNeubauer Random Variables Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example

Calculate the skewness and the kurtosis of the random variable deﬁned as the number of ﬁred bullets (see the previous examples).

Jiˇr´ıNeubauer Random Variables 3 P 3 3 µ3 = [xi − E(X )] p(xi ) = (1 − 1.56) · 0.6+ i=1 +(2 − 1.56)3 · 0.24 + (3 − 1.56)3 · 0.16 = 0.393

3 P 4 4 µ4 = [xi − E(X )] p(xi ) = (1 − 1.56) · 0.6+ i=1 +(2 − 1.56)4 · 0.24 + (3 − 1.56)4 · 0.16 = 0.756

Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example

First of all we have to calculate 3rd and 4th central moment. The mean of the given random variable is E(X ) = 1.56, the standard deviation is σ(X ) = 0.753.

Jiˇr´ıNeubauer Random Variables 3 P 4 4 µ4 = [xi − E(X )] p(xi ) = (1 − 1.56) · 0.6+ i=1 +(2 − 1.56)4 · 0.24 + (3 − 1.56)4 · 0.16 = 0.756

Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example

First of all we have to calculate 3rd and 4th central moment. The mean of the given random variable is E(X ) = 1.56, the standard deviation is σ(X ) = 0.753.

3 P 3 3 µ3 = [xi − E(X )] p(xi ) = (1 − 1.56) · 0.6+ i=1 +(2 − 1.56)3 · 0.24 + (3 − 1.56)3 · 0.16 = 0.393

Jiˇr´ıNeubauer Random Variables Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example

First of all we have to calculate 3rd and 4th central moment. The mean of the given random variable is E(X ) = 1.56, the standard deviation is σ(X ) = 0.753.

3 P 3 3 µ3 = [xi − E(X )] p(xi ) = (1 − 1.56) · 0.6+ i=1 +(2 − 1.56)3 · 0.24 + (3 − 1.56)3 · 0.16 = 0.393

3 P 4 4 µ4 = [xi − E(X )] p(xi ) = (1 − 1.56) · 0.6+ i=1 +(2 − 1.56)4 · 0.24 + (3 − 1.56)4 · 0.16 = 0.756

Jiˇr´ıNeubauer Random Variables the kurtosis is µ α = 4 − 3 = −0.644. 4 σ4

Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example

The skewness is equal to µ α = 3 = 0.922, 3 σ3

Jiˇr´ıNeubauer Random Variables Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example

The skewness is equal to µ α = 3 = 0.922, 3 σ3 the kurtosis is µ α = 4 − 3 = −0.644. 4 σ4

Jiˇr´ıNeubauer Random Variables Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example

Calculate the skewness and the kurtosis of the random X with the probability density function

12x2(1 − x) 0 < x < 1, f (x) = 0 otherwise.

Jiˇr´ıNeubauer Random Variables First of all we calculate 3rd and 4th central moment.

1 Z 2 µ = [x − 0.6]312x2(1 − x)dx = ··· = − = −0.00229, 3 875 0

1 Z 33 µ = [x − 0.6]412x2(1 − x)dx = ··· = = 0.00377. 4 8750 0

Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example

The mean of the given random variable is E(X ) = 3/5. the standard deviation is 1/5.

Jiˇr´ıNeubauer Random Variables 1 Z 33 µ = [x − 0.6]412x2(1 − x)dx = ··· = = 0.00377. 4 8750 0

Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example

The mean of the given random variable is E(X ) = 3/5. the standard deviation is 1/5. First of all we calculate 3rd and 4th central moment.

1 Z 2 µ = [x − 0.6]312x2(1 − x)dx = ··· = − = −0.00229, 3 875 0

Jiˇr´ıNeubauer Random Variables Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example

The mean of the given random variable is E(X ) = 3/5. the standard deviation is 1/5. First of all we calculate 3rd and 4th central moment.

1 Z 2 µ = [x − 0.6]312x2(1 − x)dx = ··· = − = −0.00229, 3 875 0

1 Z 33 µ = [x − 0.6]412x2(1 − x)dx = ··· = = 0.00377. 4 8750 0

Jiˇr´ıNeubauer Random Variables the kurtosis is µ 9 α = 4 − 3 = − = −0.643. 4 σ4 14

Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example

The skewness is equal to µ 2 α = 3 = − = −0.286, 3 σ3 7

Jiˇr´ıNeubauer Random Variables Discrete Random Variable Continuous Random Variable Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example

The skewness is equal to µ 2 α = 3 = − = −0.286, 3 σ3 7 the kurtosis is µ 9 α = 4 − 3 = − = −0.643. 4 σ4 14

Jiˇr´ıNeubauer Random Variables