Log-normal probability distribution A commonly used distribu- tion in probability and statistics is the log-normal distribution. (If the logarithm of a variable has a normal distribution, then the variable itself has a log-normal distribution.) The distribution function is 1 In'z/(20), for x 0, Xơ V2T f(x) where In x has zero mean and standard deviation o > 0. a. Graph f for o =5, 1, and 2. Based on your graphs, does lim f(x) appear to exist? b. Evaluate lim f(x). (Hint: Let x = e'.) c. Show that f has a single local maximum at x* = e¯". d. Evaluate f(x*) and express the result as a function of o. e. For what value of o > 0 in part (d) does f(x*) have a 2' minimum?

Log-normal probability distribution A commonly used distribu- tion in probability and statistics is the log-normal distribution. (If the logarithm of a variable has a normal distribution, then the variable itself has a log-normal distribution.) The distribution function is 1 In'z/(20), for x 0, Xơ V2T f(x) where In x has zero mean and standard deviation o > 0. a. Graph f for o =5, 1, and 2. Based on your graphs, does lim f(x) appear to exist? b. Evaluate lim f(x). (Hint: Let x = e'.) c. Show that f has a single local maximum at x* = e¯". d. Evaluate f(x) and express the result as a function of o. e. For what value of o > 0 in part (d) does f(x) have a 2' minimum?

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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