The joint pdf of the amount X of almonds and amount Y of cashews in a 1-lb can of nuts

Question

The joint pdf of the amount $X$ of almonds and amount $Y$ of cashews in a 1-lb can of nuts was:

$f(x, y) = \begin{cases} 24xy & 0 \leq x \leq 1,\, 0 \leq y \leq 1,\, x + y \leq 1\\ 0 & \text{otherwise}\end{cases}$

1. Obtain the marginal pdf of the amount $X$ of almonds in a 1-lb can.

2. What is the expected amount $E(X)$ of almonds?

Answer 1

1. By definition, we define the marginal pdf of $X$ by $$p_X(x) = \int_y f(x,y)\,dy$$ You're given that $$f(x,y) = \begin{cases}24xy & 0\leq x\leq 1,\, 0\leq y \leq 1,\, 0\leq x+y\leq 1\\ 0 & \text{otherwise}\end{cases}$$ and thus we have that $$p_X(x) = \int_?^?24xy\,dy$$ What are the appropriate limits of integration? I'll leave that to you to figure out.

2. By definition, $$E[X] = \int_x xp_X(x)\,dx$$ Thus, you can only answer this part once you've done the first part.