CDF of quotient of RVs?

Question

What would be density function for the ratio of X/Y, for both uniform and exponential distributions?

For the first one, I think that $P(C⩽c)=P(B⩾A/c)=E[e^{-A/c}]$ but unsure what that is. I am stuck on the second.

Answer 1

Here's a hint:

• Can you solve the problem if $B$ was a constant, and not a random variable?
• How can you combine the above and the law of total probability to get your answer?

Image sourced from here

Answer 2

For the first one, I think that $P(C⩽c)=P(B⩾A/c)=E[e^{-1X/z}]$ but unsure what that is.

When did random variable $X$ and constant $z$ enter into the discussion?

However, it is that: $$\begin{align}\mathsf P(C\leqslant c)&=\mathsf P(B\geqslant A/c)\\&=\mathsf E(\mathsf P(B\geqslant A/c\mid A))\\&=\mathsf E(e^{-A/c})\end{align}$$

The rest is just evaluating the expectation: $\mathsf E(g(A))=\int_\Bbb R f_A(a)~g(a)~\mathsf d a$ $$\mathsf E(e^{-A/c})=\int_0^\infty e^{-a}~e^{-a/c}~\mathsf d a$$

I am stuck on the second.

Second verse, much the same as the first.

Hint: it is a peicewise function partitioned on the magnitude of $c$.   Consider the cases $0<c< 1$ and $1\leqslant c$