I'm trying to solve the problem:
"A company will experience a loss X that is uniformly distributed between 0 and 1. The company pays a bonus to its employees that is uniformly distributed on the interval $(0, 2-X)$, which depends on the amount of the loss that occurred. Find the expected amount of the bonus paid."
Sorry, I'm really confused about how do we calculate each part of the answer.
Isn't the formula for marginal probability of $f_x(x) = \int_{-\infty}^{\infty} f(x,y) dy$. So if we don't know $f(x,y)dy$, how are we calculating $f_X(x), f_Y(y)$ or $f(y|x)$???
Best Answer
I think that the easiest approach is to use the Law of total expectation (see https://en.wikipedia.org/wiki/Law_of_total_expectation). Let $Y$ be the bonus that the employees get. By the Law of total expectation, we get that $$ \mathbb{E}[Y] = \int_{x=0}^{1} \mathbb{E}[Y \mid X = x ] \mathbb{P}(X = x) dx. $$ We know that $f(y\mid x) = 1/(2-x)$, since $Y|X=x$ is uniform on $(0,2-x)$. Using this we get that $$ \mathbb{E}[Y] = \int_{x=0}^{1} \int_{y=0}^{2-x} \frac{y}{2-x} dydx = \frac{3}{4}.$$