I'm working through an example in a textbook where the author is trying to find the PDF of the Cauchy distribution (continuous).
First they declare that $T = X/Y$ is Cauchy, where $X$ and $Y$ are both $N(0,1)$.
That leads to the definition of $T$'s CDF which we want to find as follows:
$$F_T(t)=P(X\le t|Y|)$$
Great. Now they're using LOTP to actually find the CDF. They declare an indicator random variable for the event $X \le t|Y|$, ie it's $1$ if the above equation holds true.
Then comes the part I don't get. They somehow magically go from a conditional PMF for $I$ to CDF for $t|y|$:
\begin{align}
P(I=1) & =\int_{-\infty}^\infty P(I=1\mid Y=y)f_Y(y)\,dy \\[8pt]
& = \int_{-\infty}^\infty \frac1{\sqrt{2\pi}}e^{-y^2/2} \Phi(t|y|) \, dy
\end{align}
How did they do that? I'm at the "Introduction to Probability" level, so some intuition + simple explanation would be best!
Best Answer
\begin{align} & \Pr(I=1\mid Y=y) \\[8pt] = {} & \Pr( X\le t|Y|\,\,\mid Y=y) \\[8pt] = {} & \Pr(X\le t|y|) \\[8pt] = {} & \Phi(t|y|) \end{align}