Expectation of maximum of two independent random variable with known distribution

Question

Assume $X$ and $Y$ are two random variables such that $X\sim \textrm{Unif}(0,1)$ and $Y=e^{-t}\times a $ where $t\sim \mathrm{Exp}(\lambda)$ and $a\sim \textrm{Unif}(0,1)$. What is $\mathbb{E}[\max(X,Y)]$?

Update: $X$, $t$ and $a$ are all independent random variables. By $Exp(\lambda)$, I meant the exponential distribution with mean $\frac{1}{\lambda}$.

Answer 1

A quick start, based on what seems obvious and a bit of simulation, but without doing all the steps:

If T ~ EXP(1), then exp(-T) ~ BETA(1, 1) = UNIF(0, 1), Max has support (0,1) and E(Max) = .5555.

If T ~ EXP(rate=2), then Y ~ BETA(2, 1) and E(Max) = .58.

In R (and probably about the same in Matlab):

m = 10^6; x = runif(m); a = runif(m); lam = 1

t = rexp(m, lam); y = exp(-t)*a; w = pmax(x,y)

hist(exp(-t), prob=T); mean(w)

[1] 0.5554506

I'm new to this, if partial answers and simulated clues for questions with "classical elements" are not appreciated here, then tell me so, and I will desist.