Consider two independent random variables $X$ and $Z$, where $Pr(X = 0) = Pr(X = 2) = 1/2$ and where $Z$ is uniformly distributed on $[0, 2]$. Let $Y$ be defined such that $Pr(Y = X)= 1 - Pr(Y = Z) = 1/2$
How can one start deriving the distribution function of $Y$?
For example, I understand how to do it for 1 variable. Say, let $X$ be an exponential random variable with parameter $\alpha$ then $Pr(X \leq x)= 1-\exp(-\alpha x)$
Let $Y= \alpha \exp(X)$ where $\alpha >0$. Similarly, I will try to find the corresponding cdf.
Firstly, $Y$ takes values in $(\lambda, \infty)$ since $\exp(X)$ takes values in $(1, \infty)$
Then, $Pr(Y \leq y) = Pr(\alpha \exp(X) \le y)$ for all $y \ge \lambda$
By simple transformations one arrives that the cdf is equal to $1-(\lambda/y)^\alpha$
Clearly, it is a pareto random variable.
How do I extend this logic to the case of the interaction of two variables mentioned previously?