Intuitively if you know that event a is true for a

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Unformatted text preview: one: E 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. DISCRETE-TYPE 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. 6...
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