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!

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

Recall that a moment generating function has the following form:

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

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

which is the desired answer.