What is the expected value E[X-Y] of two dependent continuous random variables with different intervals (one uniform, other exponential)?

Question

I am stuck all day with the following: Suppose I have random variable X which is exponentially distributed and random variable Y which is uniformly distributed.

$PDF_X = λexp(-λx)$ on the interval $[0,∞]$ with $E(X)=1/λ$

$PDF_Y = 1/λ$ on the interval $[0,λ]$ with $E(Y) = λ/2$

The two random variables are clearly dependent as the interval of Y is dependent on the value λ. Next to that, the intervals of the two variables do not match. I struggle hard to find $E[X-Y]$. Is it possible at all?

Apologies if this question can be found elsewhere or any mistakes. Thanks in advance for any answer.

Answer 1

Expectation is linear (see here) in the sense that $$ \operatorname E[aX+bY] =a\operatorname E[X]+b\operatorname E[Y] $$ for two random variables $X$ and $Y$ provided that the expected values of $X$ and $Y$ exist and any two constants $a$ and $b$. $X$ and $Y$ do not have to be independent or uncorrelated for the equality two hold.

We do not know if $X$ and $Y$ are dependent or independent in your example but it does not matter if we are only interested in the expected value of the difference.

I hope this helps.