find conditional probability of the new variable

Question

I am actually trying to understand the problem that how we convert our given variables into new variable and proceed them for solution in conditional probabilities, let i have two variables suppose $X_1$ and $X_2$ are iid normal $(\mu , \sigma_2^2)$ find distribution of $Y=\frac{X_1}{X_2}$

Answer 1

Ratio case

The ratio of two standard normal distribution is the Cauchy distribution. You can follow similar steps in the proof provided in this document for the non standard case.

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Log case

If $X \sim \mathcal{N}(\mu,\sigma^2)$ then $Y = \ln X$ will behave as \begin{align} f_Y(y) & = \frac {\rm d}{{\rm d}y} \Pr(Y \le y) = \frac {\rm d}{{\rm d}y} \Pr(\ln X \le y) \\[6pt] & = \frac {\rm d}{{\rm d}y} \Phi\left( \frac{e^y -\mu} \sigma \right) \\[6pt] & = \varphi\left( \frac{e^y - \mu} \sigma \right) \frac {\rm d}{{\rm d}y} \left( \frac{e^y - \mu} \sigma \right) \\[6pt] & = \varphi\left( \frac{e^y - \mu} \sigma \right) \frac {e^y} {\sigma} \\[6pt] & = \frac {e^y} {\sigma} \frac 1 {\sqrt{2\pi\,}} \exp\left( -\frac{(e^y-\mu)^2}{2\sigma^2} \right). \end{align} where $\Phi,\varphi$ are the CDF and PDF of the standard normal distribution, respectively.