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