STA642 Short Notes by Mahar Afaq

STA642 · Probability Distribution · Short Notes for Final Term · Written by Mahar Afaq Safdar Muhammadi · Note: · PMF; Probability Mass Function · PDF; Probability Density Function · CDF; Cumulative Distribution Function · MGF; Moment Generating Function · CGF; Cumulant Generating Function

Formal Definition of Independence in the case of Discrete Random Variables

Let the random variables X1 and X2 have the joint pmf p(x1,x2) and the marginal pmfs

p1(x1) and p2(x2) respectively.

The random variables X1 and X2 are said to be independent if and only if

p x x p x p x

Theorem:

If X and Y are random variables with joint distribution function F and marginal

distribution functions FX and FY, then X and Y are independent if and only if

X Y F x y F x F y

For all (x,y) ∈ R2.

Proof:

If X and Y are independent, then

X Y F x y P X x Y y P X x P Y y F x F y

Theorem:

Suppose X1 and X2 are independent and that E(u(X1)) and E(v(X2)) exist.

Then

E u X v X E u X E v X

Theorem:

Suppose the joint mgf, M(t1,t2), exists for the random variables X1 and X2.

Then, X1 and X2 are independent if and only if

,0 0, M t t M t M t

That is, the joint mgf is identically equal to the product of the marginal mgfs.

PMF of the Discrete Uniform Distribution

If x is assuming values 1,2,3,…,n, then the probability mass function (PMF) of the

discrete uniform distribution is given by

1 , 1,2,3,..., x n p x n

elsewhere

CDF of the Discrete Uniform Distribution

1, 2,..., 1, x a a a b b p X x b a

elsewhere

Mean of the Discrete Uniform Distribution

n E X

Variance of the Discrete Uniform Distribution

n E X E X

MGF of the Discrete Uniform distribution

Moment Generating Function:

n t t N n x tx tx t t x x e e M t E e e e n n n e

PMF of the Binomial distribution

Let the random variable X denote the number of successes in n trials --- when a

binomial experiment is performed.

Then the PMF of X is given by

0,1,2,..., n x x n p p x n p x x

elsewhere

where p is the probability of success in each trial and n is the number of trials.

Binomial Theorem

n n x n x

x n q p p q x

Mean of the Binomial Theorem

np

Variance of the Binomial Theorem

(1 np q npq

MGF of the Binomial distribution

The mgf of a binomial distribution is easily obtained as follows:

n n x tx tx x

x x

n x n x t

x

n t

n t n M t e p x e p p x

n e p p x

p pe

M t p pe

for all real values of t.