Mean & variance of normal distribution Let $Y = \ln(X)$, where $Y$ follows a normal distribution with mean $μ$ and variance $σ^2$. What is the mean and variance of $X$?
I know that if $Z$ follows a normal distribution with mean $μ$ and variance $σ^2$
, then $M_z(t) = e^{μt+ \tfrac{1}{2}σ^2 t^2}$

If you know that a normal distribution $Y \sim \mathcal{N}(\mu,\sigma^2)$ has a moment generating function of $M(t) = \mathbb{E}[e^{tY}] = e^{\mu t + \sigma^2t^2/2}$, then you can use that to calculate the mean and variance of $X = e^Y$ as follows:
$$\mathbb{E}[X] = \mathbb{E}[e^Y] = M(?) \cdots$$
$$\text{Var}[X] = \mathbb{E}[X^2] - \mathbb{E}[X]^2 = \mathbb{E}[e^{2Y}] - \mathbb{E}[e^Y]^2 = M(?) - M(?)^2 = \cdots$$
I'll let you fill in the $?$'s and continue the calculations from there.
