# Log Expectation [Citadel Statistics Interview Question]

Assume that log X ~ N(0, 1). What is the expectation of X?

This is the same question as problem #35 in the Statistics Chapter of Ace the Data Science Interview!

### Solution

Define $Y = \log X$. We then want to solve for:

$E[e^Y] = E[X]$

Recall that a moment generating function has the following form:

$M_Y(s) = E[e^{sY}]$

Therefore, we want the moment generating function for $Y \sim N(0,1)$, which was derived in this question and has the form:

$M_Y(s) = e^{s^2/2}$

Therefore, evaluating at $s = 1$ (since we want the mean) gives:

$M_Y(1) = e^{1/2}$

which is the desired answer.