Let $X \sim N(\mu, \sigma^2)$, where $\mu = -800$ and $\sigma = 76$
Let $Y = \exp(X)$, so Y has a lognormal distribution, $E(Y) = \exp(\mu + \sigma^2/2) = \exp(2088)$, which is a very large number.
However, if I first sample from $X$, and then take $\exp()$ of the sampled points, I got all near 0's. This is understandable since $X$ has a big negative mean and 0 is 10 standard deviations away.
But how should I understand that of $E(Y)$ being a very large number and a sample of all near 0 points? Because a very large variance?
What is the implication of the use of lognormal then?
sample <- rnorm(100000, -800, 76) summary(exp(sample)) Min. 1st Qu. Median Mean 3rd Qu. Max. 0.000e+00 0.000e+00 0.000e+00 1.170e-208 0.000e+00 1.169e-203