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Modeling and Inverse Problems in the Presence of Uncertainty
book

Modeling and Inverse Problems in the Presence of Uncertainty

by H. T. Banks, Shuhua Hu, W. Clayton Thompson
April 2014
Intermediate to advanced content levelIntermediate to advanced
405 pages
13h
English
Chapman and Hall/CRC
Content preview from Modeling and Inverse Problems in the Presence of Uncertainty
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)
kj
(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( + 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(
1
+
2
, a
2
σ
2
1
+ b
2
σ
2
2
)
for any real numbers a and b
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Publisher Resources

ISBN: 9781482206432