Fundamentals of Probability and Statistics for Engineers

Some Important Continuous Distributions 211Now, writing X ˆ ln Y , the pdf of Y can be written in the form81f Y …y† ˆ y ln Y …2† exp 1>:0; elsewhere:2 2 ln Y ln 2 y; for y 0; Y…7:47†The mean and standard deviation of Y can be found either through directintegration by using f Y (y) or by using the relationship given by Equation (7.43)together with f X (x). In terms of Y and ln Y , they take the forms m Y ˆ Y exp 2 ln Y;9> =2…7:48† 2 Y ˆ m2 Y ‰exp…2 ln Y † 1Š: > ;7.3.1 PROBABILITY TABULATIONSBecause of the close ties that exist between the normal distribution and thelognormal distribution through Equation (7.43), probability calculationsinvolving a lognormal distributed random variable can be carried out withthe aid of probability tables provided for normal random variables as shownbelow.Consider the probability distribution function of Y . We haveF Y …y† ˆP…Y y† ˆP…X ln y† ˆF X …ln y†; y 0: …7:49†2Now, since the mean of X is ln Y and its variance is ln Y,we have: ln y ln YF Y …y† ˆF U ln Y1ˆ F U ln ln Yy Since F U (u) is tabulated, Equation (7.50) can be used for probability calculationsassociated with Y with the aid of the normal probability table.Ex ample 7. 5. Problem: the annual maximum runoff Y of a certain river canbe modeled by a lognormal distribution. Suppose that the observed mean andstandard deviation of Y are m Y ˆ 300 cfs and Y ˆ 200cfs. Determine theprobability P(Y > 400 cfs).Answer: using Equations (7.48), parameters Y and ln Y are solutions of theequations Y; y 0: …7:50†TLFeBOOK