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 iswhere ln x has zero mean and standard deviation σ > 0.

a. Graph f for σ = 1/2, 1, and 2. Based on your graphs, doesappear to exist?

b. Evaluate Let x = e^y.

c. Show that f has a single local maximum at x^* = e^-σ2.

d. Evaluate f(x^*) and express the result as a function of σ.

e. For what value of σ > 0 in part (d) does f(x^*) have a minimum?

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, - In' x/(2σ), for x >0, f(x) Ξ χσV2π lim f(x) |x→0* -x1