Transcript C7_Math3033
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 ] xe 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 2Var ( X )