Consider a random variable \(X\) in the family of Poisson distributions with parameter \(\lambda\). As \(\lambda\) varies from 1 to 9 , both the mean and the variance of \(X\) increase from 1 to 9 . Now consider the transformation \(Y=\log (X+1)\). How do the mean and variance of \(Y\) vary as \(\lambda\) varies from 1 to 9 ?

Simulate 1,000 realizations of \(X\) for each \(\lambda \in\{1,2, \ldots, 9\}\), and for each value of \(\lambda\), compute the sample mean and variance of your simulated sample. How do the mean and variance vary?