A random variable that has the following pmf is said to be a binomial random variable with parameters n, p

The Binomial random variable

( ) (1 )

!where ,

( )! !

is an integer 1 and 0 1.

i n i

n i

np i p p

i

n nA

i n i i

n p

Example: A series of n independent trials, each having a probability p of being a success and 1 – p of being a failure, are performed. Let X be the number of successes in the n trials.

A random variable that has the following pmf is said to be a Poisson random variable with parameter

The Poisson random variable

( ) { } , 0,1,....!

i

p i P X i e ii

Example: The number of cars sold per day by a dealer is Poisson with parameter = 2. What is the probability of selling no cars today? What is the probability of selling 2?

Example: The number of cars sold per day by a dealer is Poisson with parameter = 2. What is the probability of selling no cars today? What is the probability of receiving 100?

Solution: P(X=0) = e-2 0.135 P(X = 2)= e-2(22 /2!) 0.270

We say that X is a continuous random variable if there exists a non-negative function f(x), for all real values of x, such that for any set B of real numbers, we have

The function f(x) is called the probability density function (pdf) of the random variable X.

Continuous random variables

{ } ( ) .B

P X B f x dx

{ ( , )} ( ) 1.

{ } ( ) ( ).

{ } ( )

{ } ( ) 0

( ) ( ) ( )

( ) ( )

a

a

b

a

a

a

a

P X f x dx

P a x a f x dx f a

P a x b f x dx

P X a f x dx

F a P x a f x dx

dF af a

da

Properties

A random variable that has the following pdf is said to be a uniform random variable over the interval (a, b)

The Uniform random variable

1 if

( )0 otherwise.

a x bf x b a

A random variable that has the following pdf is said to be a uniform random variable over the interval (a, b)

The Uniform random variable

1 if

( )0 otherwise.

0 if

( ) if

1 if .

a x bf x b a

x a

x aF x a x b

b ax b

A random variable that has the following pdf is said to be a exponential random variable with parameter

The Exponential random variable

- if 0 ( )

0 if 0

xe xf x

x

A random variable that has the following pdf is said to be a exponential random variable with parameter

The Exponential random variable

-

-

0

if 0 ( )

0 if 0

( ) 1 , 0.

x

x t x

e xf x

x

F x e dt e x

A random variable that has the following pdf is said to be a gamma random variable with parameters ,

The Gamma random variable

- 1

1

0

( ) if 0

( ) ( )

0 if 0

where ( ) .

Note: for integer , ( ) ( 1)!

x

x x

e xx

f x

x

e x dx

n n n

A random variable that has the following pdf is said to be a normal random variable with parameters ,

The Normal random variable

2 2( ) / 21( ) -

2xf x e x

Note: The distribution with parameters = 0 and = 1 is called the standard normal distribution.

If X is a discrete random variable with pmf p(x), then the expected value of X is defined by

Expectation of a random variable

[ ] ( ) ( )x x

E X xp x xP X x

If X is a discrete random variable with pmf p(x), then the expected value of X is defined by

Expectation of a random variable

[ ] ( ) ( )x x

E X xp x xP X x

Example: p(1)=0.2, p(3)=0.3, p(5)=0.2, p(7)=0.3

E[X] = 0.2(1)+0.3(3)+0.2(5)+0.3(7)=0.2+0.9+1+2.1=4.2

If X is a continuous random variable with pdf f(x), then the expected value of X is defined by

[ ] ( )E X xf x dx

Expectation of a Bernoulli random variable E[X] = 0(1 - p) + 1(p) = p

1

1 1

1[ ] ( ) (1 )n

n nE X np n n p p

p

Expectation of a Bernoulli random variable E[X] = 0(1 - p) + 1(p) = p

Expectation of a geometric random variable

1

1 1

1[ ] ( ) (1 )n

n nE X np n n p p

p

Expectation of a Bernoulli random variable E[X] = 0(1 - p) + 1(p) = p

Expectation of a geometric random variable

Expectation of a binomial random variable0 0

[ ] ( ) (1 )n n i n i

i i

nE X ip i i p p

i

1

1 1

1[ ] ( ) (1 )n

n nE X np n n p p

p

Expectation of a Bernoulli random variable E[X] = 0(1 - p) + 1(p) = p

Expectation of a geometric random variable

Expectation of a binomial random variable

Expectation of a Poisson random variable

0 0[ ] ( ) (1 )

n n i n i

i i

nE X ip i i p p

i

0 0[ ] ( )

!

i

i iE X ip i ie

i

2 2

[ ]2( ) 2

b

a

x b a a bE X dx

b a b a

Expectation of a uniform random variable

2 2

[ ]2( ) 2

b

a

x b a a bE X dx

b a b a

Expectation of a uniform random variable

Expectation of an normal random variable2 2( ) / 21

[ ]2

xE X x e

2 2

[ ]2( ) 2

b

a

x b a a bE X dx

b a b a

Expectation of a uniform random variable

Expectation of an exponential random variable

Expectation of a exponential random variable 0

1[ ] xE X x e dx

2 2( ) / 21[ ]

2xE X x e

(1) If X is a discrete random variable with pmf p(x), then for any real-valued function g,

(2) If X is a continuous random variable with pdf f(x), then for any real-valued function g,

Expectation of a function of a random variable

[ ( )] ( ) ( ) ( ) ( )x x

E g X g x p x g x P X x

[ ( )] ( ) ( )E g X g x f x dx

(1) If X is a discrete random variable with pmf p(x), then for any real-valued function g,

(2) If X is a continuous random variable with pdf f(x), then for any real-valued function g,

Expectation of a function of a random variable

[ ( )] ( ) ( ) ( ) ( )x x

E g X g x p x g x P X x

[ ( )] ( ) ( )E g X g x f x dx

Note: P(Y=g(x))=P(X=x)

If a and b are constants, then E[aX+b]=aE[X]+b

The expected value E[Xn] is called the nth moment of the random variable X.

The expected value E[(X-E[X])2] is called the variance of the random variable X and denoted by Var(X) Var(X) = E[X2] - E[X]2

Let X and Y be two random variables. The joint cumulative probability distribution of X and Y is defined as

Jointly distributed random variables

( , ) { , }, -

( ) { , } ( , )

( ) { , } ( , )

X

Y

F a b P X a Y b x

F a P X a Y F a

F a P X Y b F b

If X and Y are both discrete random variables, the joint pmf of X and Y is defined as

( , ) { , }

( ) ( , )

( ) ( , )

Xy

Yx

p x y P X x Y y

p x p x y

p y p x y

If X and Y are continuous random variables, X and Y are said to be jointly continuous if there exists a function f(x, y) such that

( , ) ( , )A B

P x A y B f x y dxdy

{ } { , ( , )} ( , )

{ } ( , ) ( )

( ) ( , )

( ) ( , )

A

XA

X

Y

P X A P X A Y f x y dxdy

P X A f x y dxdy f x dx

f x f x y dy

f y f x y dx