Fundamentals of Probability and Statistics for Engineers

Expectations and Moments 81f X (x)σ1xσ 2 > σ 2 1 Figure 4.2 Density functions with different variances, ,and 2We note two other properties of the variance of a random variable X whichcan be similarly verified. They are:)var…X ‡ c† ˆvar…X†;…4:9†var…cX† ˆc 2 var…X†;where c is any constant.It is further noted from Equations (4.6) and (4.7) that, since each term in thesum in Equation (4.6) and the integrand in Equation (4.7) are nonnegative, thevariance of a random variable is always nonnegative. The positive square root X ˆ‡‰Ef…X m† 2 gŠ 1=2 ;is called the standard deviation of X. An advantage of using X rather than 2 Xas a measure of dispersion is that it has the same unit as the mean. It cantherefore be compared with the mean on the same scale to gain some measureof the degree of spread of the distribution. A dimensionless number thatcharacterizes dispersion relative to the mean which also facilitates comparisonamong random variables of different units is the coefficient of variation, v X ,defined byv X ˆ Xm X:…4:10†Ex ample 4. 5. Let us determine the variance of Y defined in Example 4.1.Using Equation (4.8), we may write 2 Y ˆ EfY 2 g m 2 Y ˆ EfY 2 g n 2 q 2 :TLFeBOOK