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The Binomial random variable
A random variable that has the following pmf is said to be a
binomial random variable with parameters n, p
n i
p(i )    p (1  p) n i
i
n
n!
where    n Ai 
,
(n  i )!i !
i
n is an integer  1 and 0  p  1.
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.
The Poisson random variable
A random variable that has the following pmf is said to be a
Poisson random variable with parameter l (l > 0)
p(i)  P{ X  i}  e l
li
i!
, i  0,1,....
Example: The number of cars sold per day by a dealer is
Poisson with parameter l = 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 l = 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
Continuous random variables
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
P{ X  B}   f ( x)dx.
B
The function f(x) is called the probability density function
(pdf) of the random variable X.
Properties

 P{ X  (, )}  

f ( x)dx  1.
 P{a    x  a   }  
a 
a 
f ( x )dx   f (a ).
b
 P{a  x  b}   f ( x)dx
a
a
 P{ X  a}   f ( x)dx  0
a
 F (a)  P( x  a )  
a

dF (a )

 f (a)
da
f ( x )dx
The Uniform random variable
A random variable that has the following pdf is said to be a
uniform random variable over the interval (a, b)
 1

f ( x)   b  a

0
if a  x  b
otherwise.
The Uniform random variable
A random variable that has the following pdf is said to be a
uniform random variable over the interval (a, b)
 1

f ( x)   b  a

0
0
 xa

F ( x)  
 ba

 1
if a  x  b
otherwise.
if x  a
if a  x  b
if x  b.
The Exponential random variable
A random variable that has the following pdf is said to be a
exponential random variable with parameter l > 0
l e - l x
if x  0
f ( x)  
if x  0
0
The Exponential random variable
A random variable that has the following pdf is said to be a
exponential random variable with parameter l > 0
l e - l x
f ( x)  
0
x
if x  0
if x  0
F ( x)   l e- lt dt  1  e  l x ,
0
x  0.
The Gamma random variable
A random variable that has the following pdf is said to be a
gamma random variable with parameters a, l > 0
 l e- l x (l x)a 1

f ( x)  
(a )
0

if x  0
if x  0
x
where (a )   e  x xa 1dx.
0
Note: for integer n, ( n)  ( n  1)!
The Normal random variable
A random variable that has the following pdf is said to be a
normal random variable with parameters m, s2
f ( x) 
1
 ( x  m )2 / 2s 2
e
2s
-  x  
Note: The distribution with parameters m = 0 and s = 1
is called the standard normal distribution.
Expectation of a random variable
If X is a discrete random variable with pmf p(x), then the
expected value of X is defined by
E[ X ]   xp( x)   xP( X  x)
x
x
Expectation of a random variable
If X is a discrete random variable with pmf p(x), then the
expected value of X is defined by
E[ X ]   xp( x)   xP( X  x)
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
Expectation of a Bernoulli random variable
E[X] = 0(1 - p) + 1(p) = p
Expectation of a geometric random variable
1


n 1
E[ X ]   n 1 np(n)  n 1 n(1  p) p 
p
Expectation of a Bernoulli random variable
E[X] = 0(1 - p) + 1(p) = p
Expectation of a geometric random variable
1


n 1
E[ X ]   n 1 np(n)  n 1 n(1  p) p 
p
Expectation of a binomial random variable
n i
n
n
E[ X ]   i 0 ip(i )  i 0 i   p (1  p) n i
i
Expectation of a Bernoulli random variable
E[X] = 0(1 - p) + 1(p) = p
Expectation of a geometric random variable
1


n 1
E[ X ]   n 1 np(n)  n 1 n(1  p) p 
p
Expectation of a binomial random variable
n i
n
n
E[ X ]   i 0 ip(i )  i 0 i   p (1  p) n i
i
Expectation of a Poisson random variable
E[ X ]   i 0 ip(i )  i 0 ie l


li
i!
l
Expectation of a uniform random variable
b
x
b2  a 2 a  b
E[ X ]  
dx 

a ba
2(b  a)
2
Expectation of a uniform random variable
b
x
b2  a 2 a  b
E[ X ]  
dx 

a ba
2(b  a)
2
Expectation of an normal random variable

1
 ( x  m )2 / 2s 2
E[ X ]   x
e
m

2s
Expectation of a uniform random variable
b
x
b2  a 2 a  b
E[ X ]  
dx 

a ba
2(b  a)
2
Expectation of an exponential random variable

1
 ( x  m )2 / 2s 2
E[ X ]   x
e
m

2s
Expectation of a exponential random variable

1
l x
E[ X ]   xl e dx 
0
l
Expectation of a function
of a random variable
(1) If X is a discrete random variable with pmf p(x), then for
any real-valued function g,
E[ g ( X )]   g ( x) p( x)   g ( x) P ( X  x)
x
x
(2) If X is a continuous random variable with pdf f(x), then
for any real-valued function g,

E[ g ( X )]   g ( x) f ( x)dx

Expectation of a function
of a random variable
(1) If X is a discrete random variable with pmf p(x), then for
any real-valued function g,
E[ g ( X )]   g ( x) p( x)   g ( x) P ( X  x)
x
x
(2) If X is a continuous random variable with pdf f(x), then
for any real-valued function g,

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
Jointly distributed random variables
Let X and Y be two random variables. The joint cumulative
probability distribution of X and Y is defined as
F (a, b)  P{ X  a, Y  b},
-  x  
FX (a )  P{ X  a, Y  }  F (a, )
FY (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
p ( x, y )  P{ X  x, Y  y}
p X ( x )   p ( x, y )
y
pY ( y )   p ( x, y )
x
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
P( x  A, y  B)    f ( x, y )dxdy
A B
P{ X  A}  P{ X  A, Y  (, )}  
P{ X  A}  
f X ( x)  

fY ( y )  





A 
f ( x, y )dxdy  
f ( x, y )dy
f ( x, y )dx




A 
f ( x, y )dxdy
f X ( x)dx