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Unformatted text preview: how spread out 2 the distribution of X is. As before, the variance of X is often denoted by σX , where σX = Var(X ) is called the standard deviation of X . If X is a measurement in some units, then σX is in the same units. For example. if X represents a measurement in feet, then Var(X ) represents a number of feet2 and σX represents a number of feet. Exactly as shown in Section 2.2 for discrete-type random variables, the variance for continuous-type random variables scales as Var(aX + b) = a2 Var(X ). −µ The standardized random variable, XσX X , is a dimensionless random variable with mean zero and variance one. Finally, as before, the variance of a random variable is equal to its second moment minus the square of its first moment: Var(X ) = E [X 2 − 2µX X + µ2 ] X = E [X 2 ] − 2µX E [X ] + µ2 X = E [X 2 ] − µ2 . X Example 3.2.2 Suppose X has the following pdf, where A is a constant to be determined: fX (u) = A(1 − u2 ) −1 ≤ u ≤ 1 0 else. Find A, P {0.5 < X < 1.5}, FX , µX , Var(X ), and σX . 76 Solution: CHAPTER 3. CONTINUOUS-TYPE RANDOM VARIABLES To find A we require that ∞ 1...
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