If X1 is a lognormal random variable with parameters (α, β1 ), and X2 is a lognormal random variable with parameters (α, β2 ), it has been postulated that the product: Y = X1 X2 has a lognormal distribution with parameters (α, βy ) where: βy = β1 + β2

(i) Con?rm this result theoretically. (Hint : use results from Chapter 6 regarding the sum of Gaussian random variables.)

(ii) In the data table below, X1 is a random sample drawn from a distribution pur- ported to be L(0.25, 0.50); and X2 is a random sample drawn from a distribution purported to be L(0.25, 0.25).

From this data set, obtain the corresponding values for Y de?ned as the product Y = X1 X2 . According to the result stated and proved in (i), what is the theoretical distribution of Y ? Con?rm that the computed sample data set for Y agrees with this postulate.