If X ∼ N(µ, σ2 ) find the pdf of Y = e ^ X.

Question

Help? I am stuck on this homework question and finding very difficult to answer

Answer 1

An approach (outline): you can for instance go through the cumulative distribution function, then differentiate it to get the pdf ($f_Y(x) = F^\prime_Y(x)$). For $x \in\mathbb{R}$, $$F_Y(x) = \mathbb{P}\{ Y \leq x \} = \mathbb{P}\{ e^X \leq x \}.$$

If $ x \leq 0$, this probability is $0$ as $e^X \geq 0$ a.s. For $x > 0$, you get

$$F_Y(x) = \mathbb{P}\{ e^X \leq x \}= \mathbb{P}\{ X \leq \ln x \} = F_X(\ln x)$$ which you know explicitly (as $X$ is Gaussian with known parameters).