Let $T = \frac{X}{S}$ with X a random exponential variable of parameter lambda and S gamma random variable of this form: $\Gamma(2,\lambda)$.
I want to find the density of T. Know, I tried to find $F_T(t)$, so that I can derive and obtain the desired result. This is what I did:
$$ F_T(t) = \mathbb{P}[T \leq t] = \mathbb{P}[\frac{X}{S} \leq t]$$
Then I am not sure if I should write S = X + Y with X and Y independent random variable of exponential distribution $\lambda$ or if I should continue with the definition.
I tried to keep going with the definition:
$$ \mathbb{P}[\frac{X}{S} \leq t] = \mathbb{P}[X \leq tS] $$ However now I cannot say that this is equal to $F_X(tS)$ because I have S as random variable, so I tried to write it in this way: $$\mathbb{P}[X \leq ts \ \vert S = s]$$ So that I can use Bayes theorem and write: $$\frac{\mathbb{P}[X \leq ts \cap S = s]}{\mathbb{P}[S = s]}$$ However here I am not sure on how to proceed. Any idea?
