?Log-normal probability distribution A commonly used distribution in probability and

Chapter 3, Problem 452

(choose chapter or problem)

Log-normal probability distribution A commonly used distribution 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

\(f(x)=\frac{1}{x \sigma \sqrt{2 \pi}} e^{-\ln ^{2} x /\left(2 \sigma^{2}\right)}\), for x \(\geq\) 0

where In x has zero mean and standard deviation (\sigma\) > 0.

a. Graph f for \(\sigma=\frac{1}{2}, 1,2\). Based on your graphs, does \(\lim _{x \rightarrow 0} f(x)\) appear to exist?

b. Evaluate \(\left.\lim _{x \rightarrow 0} f(x) \). (Hint: Let \( x=e^{y}\).)

c. Show that f has a single local maximum at \(x^{*}=e^{-\sigma^{2}}\)

d. Evaluate f(\(x^{*}) and express the result as a function of \(\sigma\).

e. For what value of \(\sigma\)> in part (d) does f(\(x^{*}\)) have a minimum?