
Probability and Statistics Overview 33
and the kth moments are given in terms of the c entral moments by
E(X
k
) =
k
X
j=0
k
j
µ
j
E
(X −µ)
k−j
(2.83)
where
k
j
=
k!
j!(k − j)!
.
The normal distribution ha s a number of useful properties . For example, it
is closed under linear transformation. In other words, if X ∼ N(µ, σ
2
), then
aX + b ∼ N(aµ + b, a
2
σ
2
)
for any real numbers a and b. This implies that if X ∼ N(µ, σ
2
), then
X −µ
σ
∼ N(0, 1).
In addition, if X
1
∼ N(µ
1
, σ
2
1
) and X
2
∼ N(µ
2
, σ
2
2
) are independent, then any
linear combination of thes e two random variables is also normally distributed,
that is,
aX
1
+ bX
2
∼ N(aµ
1
+ bµ
2
, a
2
σ
2
1
+ b
2
σ
2
2
)
for any real numbers a and b