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Moment generating function
The moment generating function of random variable X is
given by
 (t )  E[etX ]
Moment generating function
The moment generating function of random variable X is
given by
 (t )  E[etX ]
 (t )   etx p( x) if X is discrete
x
Moment generating function
The moment generating function of random variable X is
given by
 (t )  E[etX ]
 (t )   etx p( x) if X is discrete
x

 (t )   etx f ( x)dx if X is continuous

dE[etX ]
d tX
 '(t ) 
 E[ e ]  E[ XetX ]
dt
dt
  '(0)  E[ X ]
dE[etX ]
d tX
 '(t ) 
 E[ e ]  E[ XetX ]
dt
dt
  '(0)  E[ X ]
d
 (t )  E[ XetX ]  E[ X 2 etX ]
dt
  (2) (0)  E[ X 2 ]
(2)
More generally,
d k 1 tX
 (t )  E[ X e ]  E[ X k etX ]
dt
  ( k ) (0)  E[ X k ]
(k )
Example: X has the Poisson distribution with parameter l
Example: X has the Poisson distribution with parameter l
x l
x l
(
l
)
e
(
l
)
e

tX
tx
tx
 (t )  E[e ]   x 0 e
  x 0 e
x!
x!
t x
t
(
l
e
)

l
 l l et
l ( ee 1)
=e  x 0
e e e
x!

Example: X has the Poisson distribution with parameter l
x l
x l
(
l
)
e
(
l
)
e

tX
tx
tx
 (t )  E[e ]   x 0 e
  x 0 e
x!
x!
t x
(
l
e
)

l
 l l et
l ( et 1)
=e  x 0
e e e
x!

  (t ) 
d [e
l ( et 1)
dt
]
 le e
t l ( et 1)
Example: X has the Poisson distribution with parameter l
x l
x l
(
l
)
e
(
l
)
e

tX
tx
tx
 (t )  E[e ]   x 0 e
  x 0 e
x!
x!
t x
(
l
e
)

l
 l l et
l ( et 1)
=e  x 0
e e e
x!

  (t ) 
d [e
l ( et 1)
dt
]
 l et el ( e 1)
t
ax
a

e
 x 0 x!

t l ( et 1)
 '(0)  l e e
t 0
l
t l ( et 1)
 '(0)  l e e
t l ( et 1)
 ''(0)  l e e
t 0
l
t l ( et 1)
 le le e
t
t 0
 l  l2
 '(0)  l e e
t l ( et 1)
t l ( et 1)
 ''(0)  l e e
t 0
l
t l ( et 1)
 le le e
t
 Var ( X )  E[ X 2 ]  E[ X ]2  l
t 0
 l  l2
If X and Y are independent, then
 X Y (t )  E[et ( X Y ) ]  E[etX etY ) ]  E[etX ]E[etY ]
= X (t )Y (t )
The moment generating function of the sum of two random
variables is the product of the individual moment generating
functions
Let Y = X1+X2 where X1~Poisson(l1) and X2~Poisson(l2) and
X1 and X1 are independent, then
E[etY ]  E[et ( X1  X 2 ) ]  E[etX1 ]E[etX 2 ]
=e
l1 ( et 1) l2 ( et 1)
e
e
( et 1)( l1  l2 )
Let Y = X1+X2 where X1~Poisson(l1) and X2~Poisson(l2) and
X1 and X1 are independent, then
E[etY ]  E[et ( X1  X 2 ) ]  E[etX1 ]E[etX 2 ]
=e
l1 ( et 1) l2 ( et 1)
e
 Y ~ Poisson(l1  l2 )
e
( et 1)( l1  l2 )
Note: The moment generating function uniquely determines
the distribution.
Markov’s inequality
If X is a random variable that takes only nonnegative values,
then for any a > 0,
E[ X ]
P( X  a) 
.
a
Proof (in the case where X is continuous):

E[ X ]   xf ( x)dx

a


a
  xf ( x)dx   xf ( x)dx

  xf ( x)dx
a


a
a
  af ( x)dx  a  f ( x)dx  aP( X  a)
Strong law of large numbers
Let X1, X2, ..., Xn be a set of independent random variables
having a common distribution, and let E[Xi] = m. then, with
probability 1
X1  X1  ...  X n
 m as n  .
n
Central Limit Theorem
Let X1, X2, ..., Xn be a set of independent random variables
having a common distribution with mean m and variance s.
Then the distribution of
X 1  X 1  ...  X n  nm
s n
tends to the standard normal as n  . That is
P(
X 1  X 1  ...  X n  nm
as n  .
s n
 a) 
1
2

a

e
 x2 / 2
dx
Conditional probability and
conditional expectations
Let X and Y be two discrete random variables, then the
conditional probability mass function of X given that Y=y is
defined as
P{ X  x, Y  y} p( x, y)
pX |Y ( x | y)  P{ X  x | Y  y} 

.
P{Y  y}
p( y )
for all values of y for which P(Y=y)>0.
Conditional probability and
conditional expectations
Let X and Y be two discrete random variables, then the
conditional probability mass function of X given that Y=y is
defined as
P{ X  x, Y  y} p( x, y)
pX |Y ( x | y)  P{ X  x | Y  y} 

.
P{Y  y}
p( y )
for all values of y for which P(Y=y)>0.
The conditional expectation of X given that Y=y is defined as
E[ X | Y  y ]   xP{ X  x | Y  y}   xp X |Y ( x | y ).
x
x
Let X and Y be two continuous random variables, then the
conditional probability density function of X given that Y=y
is defined as
f ( x, y)
f X |Y ( x | y) 
.
fY ( y )
for all values of y for which fY(y)>0.
The conditional expectation of X given that Y=y is defined as

E[ X | Y  y]   xf X |Y ( x | y)dx.

E[ X ]  E[ E[ X | Y  y ]]  E[ E[ X | Y ]]
E[ X ]   E[ X | Y  y ]P(Y  y ) if Y is discrete
y

E[ X ]   E[ X | Y  y ] f ( y )dy if Y is continuous.

Proof: