Question

LetXandYbe independent random variables suchthat log(X)has the normal distribution with mean 1.6 andvariance 4.5 and log(Y )has the normal distribution withmean 3 and variance 6. Find the distribution of the productXY.

Short answer

The product of XY has the lognormal distribution with mean 4.6 and variance 10.5.

Step-by-step solution

Given information

X and Y both are independent lognormal random variables.

\[\begin{array}{l}\log \left( X \right) \sim N\left( {1.6,4.5} \right)\\\log \left( Y \right) \sim N\left( {3,6} \right)\end{array}\]

Determine the distribution

Since X and Y both are independent.

\[\log \left( {XY} \right) = \log \left( X \right) + \log \left( Y \right)\]

Since X and Y follow a lognormal distribution, the product of XY also follows a lognormal distribution.

Hence,

\[\begin{array}{l}\log \left( {XY} \right) \sim N\left( {1.6 + 3,4.5 + 6} \right)\\\log \left( {XY} \right) \sim N\left( {4.6,10.5} \right)\end{array}\]

Therefore,the product of XY has the lognormal distribution with mean 4.6 and variance 10.5.