Fundamentals of Probability and Statistics for Engineers

210 Fundamentals of Probability and Statistics for Engineers1.5f Y (y)1.0σ2X = 0.1σ2X = 0.30.5σ2X = 1.00.00 2 4 6 8yFigure 7. 9 Lognormal distribution, f Y (y), with m X ˆ 0, for several values of 2 Xquantities that are restricted to having only positive values. In addition, f Y (y)takes many different shapes for different values of m X and X ( x > 0). As seenfrom Figure 7.9, the pdf of Y is skewed to the right, this characteristic becomingmore pronounced as X increases.It is noted that parameters m X and X appearing in the pdf of Y are themean and standard deviation of X, or ln Y , but not of Y . To obtain a morenatural pair of parameters for f Y (y), we observe that, if medians of X and Y aredenoted by X and Y , respectively, the definition of the median of a randomvariable gives0:5 ˆ P…Y Y †ˆP…X ln Y †ˆP…X X †;orln Y ˆ X :…7:45†Since, owing to the symmetry of the normal distribution,we can write X ˆ m X ;m X ˆ ln Y :…7:46†TLFeBOOK