I want to come up with at least the expectation, and at best, the cdf, for a variable $Z$ that I think of as the result of a process and am not quite sure how to translate into equations.
Let $F(x) = x$, where $F(x)$ is the cdf defined over $(0,1)$ of random variable $X$ (uniform distribution).
Now let $G(y) = f1(F(y))$ where $f1$ in this case is just some function (I figure I don't need to write out the whole thing), with $Y\sim G(y)$, [EDIT: and $G$ is a valid cdf ($f1$ is a specific function that preserves cdf properties, I'm just using $f1$ for shorthand so I don't complicate the question with a long complicated function)].
Similarly, $H(w) = f2(F(w))$ and $W\sim H(w)$, [EDIT: and $H$ is a valid cdf ($f2$ is a specific function that preserves cdf properties, I'm just using $f2$ for shorthand so I don't complicate the question with a long complicated function)]
Now, I want to define a random variable $Z$ such that $Z$ is a weighted average of either a draw from $G$ (with probability $a$) or a draw from $H$ that is strictly greater than whatever the draw from $G$ was (with probability $1-a$).
So, in other words, I want to write out the distribution of $Z$ where $Z$ looks something like (I know this is not quite correct form) $Z= a Y + (1-a)(W|W>Y)$
Can anyone help me out?
Thanks so much!