the moment generating function of random variable x is given by moment generating function
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The moment generating function of random variable X is given by
Moment generating function
( ) [ ]tXt E e
The moment generating function of random variable X is given by
Moment generating function
( ) [ ]
( ) ( ) if is discrete
tX
tx
x
t E e
t e p x X
The moment generating function of random variable X is given by
Moment generating function
( ) [ ]
( ) ( ) if is discrete
( ) ( ) if is continuous
tX
tx
x
tx
t E e
t e p x X
t e f x dx X
[ ]'( ) [ ] [ ]
'(0) [ ]
tXtX tXdE e d
t E e E Xedt dtE X
(2) 2
(2) 2
[ ]'( ) [ ] [ ]
'(0) [ ]
( ) [ ] [ ]
(0) [ ]
tXtX tX
tX tX
dE e dt E e E Xe
dt dtE X
dt E Xe E X e
dt
E X
( ) 1
( )
( ) [ ] [ ]
(0) [ ]
k k tX k tX
k k
dt E X e E X e
dt
E X
More generally,
Example: X has the Poisson distribution with parameter
0 0
( 1)
0
( ) ( )( ) [ ]
! !
( ) =
!
tt e
x xtX tx tx
x x
t xe e
x
e et E e e e
x x
ee e e e
x
Example: X has the Poisson distribution with parameter
0 0
( 1)
0
( 1)( 1)
( ) ( )( ) [ ]
! !
( ) =
!
[ ]( )
t t
t
t
x xtX tx tx
x x
t xe e
x
et e
e et E e e e
x x
ee e e e
x
d et e e
dt
Example: X has the Poisson distribution with parameter
0 0
( 1)
0
( 1)( 1)
( ) ( )( ) [ ]
! !
( ) =
!
[ ]( )
t t
t
t
x xtX tx tx
x x
t xe e
x
et e
e et E e e e
x x
ee e e e
x
d et e e
dt
Example: X has the Poisson distribution with parameter
0 !
xa
x
ae
x
( 1)
0'(0)
tt e
te e
( 1)
0
( 1) ( 1) 2
0
'(0)
''(0)
t
t t
t e
t
t e t t e
t
e e
e e e e e
( 1)
0
( 1) ( 1) 2
0
2 2
'(0)
''(0)
( ) [ ] [ ]
t
t t
t e
t
t e t t e
t
e e
e e e e e
Var X E X E X
If X and Y are independent, then( ) )( ) [ ] [ ] [ ] [ ]
= ( ) ( )
t X Y tX tY tX tYX Y
X Y
t E e E e e E e E e
t 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(1) and X2~Poisson(2) and X1 and X1 are independent, then
1 2 1 2
1 2 1 2
( )
( 1) ( 1) ( 1)( )
[ ] [ ] [ ] [ ]
=t t t
t X X tX tXtY
e e e
E e E e E e E e
e e e
Let Y = X1+X2 where X1~Poisson(1) and X2~Poisson(2) and X1 and X1 are independent, then
1 2 1 2
1 2 1 2
( )
( 1) ( 1) ( 1)( )
1 2
[ ] [ ] [ ] [ ]
=
~ Poisson( )
t t t
t X X tX tXtY
e e e
E e E e E e E e
e e e
Y
Note: The moment generating function uniquely determines the distribution.
If X is a random variable that takes only nonnegative values, then for any a > 0,
Markov’s inequality
[ ]( ) .
E XP X a
a
[ ] ( )
( ) ( )
( )
( ) ( ) ( )
a
a
a
a a
E X xf x dx
xf x dx xf x dx
xf x dx
af x dx a f x dx aP X a
Proof (in the case where X is continuous):
Let X1, X2, ..., Xn be a set of independent random variables having a common distribution, and let E[Xi] = . then, with probability 1
1 1 ... as .nX X X
nn
Strong law of large numbers
Let X1, X2, ..., Xn be a set of independent random variables having a common distribution with mean and variance Then the distribution of
2
1 1
/ 21 1
...
tends to the standard normal as . That is
... 1( )
2as .
n
a xn
X X X n
nn
X X X nP a e dx
nn
Central Limit Theorem
Let X and Y be two discrete random variables, then the conditional probability mass function of X given that Y=y is defined as
for all values of y for which P(Y=y)>0.
|
{ , } ( , )( | ) { | } .
{ } ( )X Y
P X x Y y p x yp x y P X x Y y
P Y y p y
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
for all values of y for which P(Y=y)>0.
The conditional expectation of X given that Y=y is defined as
|
{ , } ( , )( | ) { | } .
{ } ( )X Y
P X x Y y p x yp x y P X x Y y
P Y y p y
Conditional probability and conditional expectations
|[ | ] { | } ( | ).X Yx x
E X Y y xP X x Y y xp x y
Let X and Y be two continuous random variables, then the conditional probability density function of X given that Y=y is defined as
for all values of y for which fY(y)>0.
The conditional expectation of X given that Y=y is defined as
|
( , )( | ) .
( )X YY
f x yf x y
f y
|[ | ] ( | ) .X YE X Y y xf x y dx
[ ] [ [ | ]] [ [ | ]]
[ ] [ | ] ( ) if is discrete
[ ] [ | ] ( ) if is continuous.
y
E X E E X Y y E E X Y
E X E X Y y P Y y Y
E X E X Y y f y dy Y
Proof: