A random variable Y is said to have a lognormal distribution if log Y has a normal distribution. Equivalently, we can write Y = e^X, where X has a normal distribution.

(a) If X₁, X₂, . . . is an independent sequence of uniform (0, 1) variables, show that the product has an approximate lognormal distribution. Show that the mean and variance of log Y are, respectively, ˆ’n and n.

(b) If Y = e^X, with X ˆ¼ Norm(μ, σ²), it can be shown that

Let X₁, . . . , X₁₀₀ be an independent sequence of uniform (0,1) variables. Estimate

(c) Verify the aforementioned results with a simulation experiment in R.

Y = II-1 X; n. i i=1 E[Y] = e#+o/2 and V[Y] = (e*-1) e e2H+o