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MATH 3033 based on
Dekking et al. A Modern Introduction to Probability and Statistics. 2007
Slides by Yu Liang
Instructor Longin Jan Latecki
Chapter 7: Expectation and variance
The expectation of a discrete random variable X taking the values a1,
a2, . . . and with probability mass function p is the number:
E[ X ]   ai P( X  ai )   ai p(ai )
i
i
We also call E[X] the expected value or mean of X. Since the expectation is
determined by the probability distribution of X only, we also speak of the
expectation or mean of the distribution.
Expected values
of discrete random variable
The expectation of a continuous random variable X with probability density
function f is the number

E[ X ] 
 xf ( x)dx

We also call E[X] the expected value or mean of X. Note that E[X] is indeed
the center of gravity of the mass distribution described by the function f:


E[ X ] 
 xf ( x)dx 

 xf ( x)dx
-

 f ( x)dx
-
Expected values
of continuous random variable
The EXPECTATION of a GEOMETRIC DISTRIBUTION. Let X have a
geometric distribution with parameter p; then

E[ X ]   kp(1  p )k 1 
k 1
1
p
The EXPECTATION of an EXPONENTIAL DISTRIBUTION. Let X have an
exponential distribution with parameter λ; then

E[ X ]   xe dx 
x
0
1

The EXPECTATION of a GEOMETRIC DISTRIBUTION. Let X have a
geometric distribution with parameter p; then

1 X 
 (
1
E[ X ]   x
e 2
  2

)2
dx  
The CHANGE-OF-VARIABLE FORMULA. Let X be a random variable, and
let g : R → R be a function.
If X is discrete, taking the values a1, a2, . . . , then
E[ g ( X )]   g (ai )P( X  ai )
i
If X is continuous, with probability density function f, then

E[ g ( X )] 
 g ( x) f ( x)dx

The variance Var(X) of a random variable X is the number
Var ( X )  E[( X  E[ X ]) 2 ]
Standard deviation: Var ( X )
Variance of a normal distribution. Let X be an N(μ, σ2) distributed random
variable. Then

1 x 
 (
1
2
Var ( X )   ( x   )
e 2
 2


)2
dx   2
An alternative expression for the variance. For any random variable X,
Var ( X )  E[ X 2 ]  E[ X ]2
E[ X 2 ] is called the second moment of X. We can derive this equation from:


Var ( X )  E[( X  E[ X ]2 ]   ( x  E[ X ]2 f ( x )dx   ( x 2  2 xE[ X ]  ( E[ X ]) 2 ) f ( x )dx



x

2





f ( x )dx  2 E[ X ]  xf ( x )dx  ( E[ X ]) 2  f ( x )dx  E[ X 2 ]  2( E[ X ]) 2  ( E[ X ]) 2
 ( E[ X ])  ( E[ X ])
2
2
Expectation and variance under change of units. For any random variable X
and any real numbers r and s,
E[rX  s ]  rE[ X ]  s
and
Var ( rX  s)  r 2 Var ( X )