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).
|
X1
1.16741
|
X2
1.61889
|
|
1.58631
|
1.15897
|
|
2.00530
|
1.17163
|
|
1.67186
|
1.09065
|
|
1.63146
|
1.27686
|
|
1.61738
|
0.91838
|
|
0.74154
|
1.45123
|
|
2.96673
|
1.47800
|
|
1.50267
|
2.16068
|
|
1.99272
|
1.46116
|
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.