THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
STAT1301 PROBABILITY AND STATISTICS I
EXAMPLE CLASS 5
Review
1. Expectation
Assume that
E
(
X
) and
E
(
g
(
X
)) exist in the following, then
E
(
X
)
E
(
g
(
X
))
X
is a discrete r.v.
∑
Ω
∈
)
(
)
(
X
x
x
xp
∑
Ω
∈
)
(
)
(
)
(
X
x
x
p
x
g
X
is a continuous r.v.
∫
Ω
)
(
)
(
X
dx
x
xf
∫
Ω
)
(
)
(
)
(
X
dx
x
f
x
g
Mean of
X
=
E
(
X
)
Mean and variance
Variance of
X
=
Var
(
X
) =
(
)
2
))
(
(
X
E
X
E
−
=
E
(
X
2
) –
E
(
X
)
2
The mean is usually denoted by
µ
and the variance is usually denoted by
σ
2
.
2. Moments and moment generating function
Let
r
be a positive integer. The
r
th moment of
X
is
E
(
X
r
). The
r
th moment of
X
about
b
is
Moment
(
)
r
b
X
E
)
(
−
.
Let
X
be a random variable. The moment generating function of
X
is defined as
Moment generating function
M
X
(
t
) =
E
(
e
tX
)
if it exists. The domain of
𝑀
𝑋
(
𝑡
)
are all real number
t
such that the expectation
above is finite.
And we have
𝑀
𝑋
(
𝑟
)
(0) =
𝜕
𝑟
𝑀
𝑋
(
𝑡
)
𝜕𝑡
𝑟
�
𝑡=0
=
E
(
X
r
)
if
M
X
(
t
) is differentiable at
t
= 0.
Remark: moment generating function uniquely characterizes the distribution.
3. Some common continuous distributions
Uniform distribution.
Let
X
~
U
(
a
,
b
), where
b
>
a
. The probability density function
(pdf) of
X
is

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