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Var X − µX
σX
X − µX
σX =
= 1
(E [X ] − µX ) = 0
σX
σ2
1
E [(X − µX )2 ] = X = 1.
2
2
σX
σX Note that even if X is a measurement in some units such as meters, then the standardized random
−µ
variable XσX X is dimensionless, because the standard deviation σX is in the same units as X.
Using the linearity of expectation, we can ﬁnd another expression for Var(X ) which is often
more convenient for computation than the deﬁnition of variance itself:
Var(X ) = E [X 2 − 2µX X + µ2 ]
X
= E [X 2 ] − 2µX E [X ] + µ2
X
= E [X 2 ] − µ2 .
X
For an integer i ≥ 1, the ith moment of X is deﬁned to be E [X i ]. Therefore, the variance of a
random variable is equal to its second moment minus the square of its ﬁrst moment. 28 CHAPTER 2. DISCRETETYPE RANDOM VARIABLES Example 2.2.7 Let X denote the number showing for one roll of a fair die. Find Var(X ) and the
standard deviation, σX .
Solution: As noted in Example 2.1.2, µX = 3.5. Also, E [X 2 ] =
Var(X ) = 91 − (3.5)2 ≈ 2.9167 and σX = Var(X ) = 1.7078.
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 Spring '08
 Zahrn
 The Land

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