# Suppose the joint pdf of 𝑿 and 𝒀 answer the following questions: [duplicate]

How can I answer the questions with these information

• X and Y are not independent. For example if X=30, Y must =0. Statement has f(x). Shouldn't it be f(x,y)? Apr 7, 2021 at 21:59
• Yes it should be f(x,y) Apr 7, 2021 at 22:17

Analytic geometry is the way to approach this problem. The area of interest is a trapezoid $$T$$ with bounds positive $$x$$ axis, positive $$y$$ axis, and lines $$x+y=20$$ and $$x+y=30$$. The first step is to calculate $$k$$ so that $$\int_Tf(x,y)dxdy=1$$. To get $$P(X+Y\le 25)$$, draw line $$x+y=25$$ and integrate $$f(x,y)$$ over that part of $$T$$ to the left of that line.
$$E(X+Y)=\int_T(x+y)f(x,y)dxdy=m$$. To get $$V(X+Y)$$, get second moment $$M= \int_T(x+y)^2f(x,y)dxdy$$ and $$V(X+Y)=M-m^2$$
For the last two, you need $$E(XY)$$, $$E(X^k)$$ and $$E(Y^k)$$ for $$k=1$$ and $$k=2$$ calculated the same way as other expectations. Standard formulas apply.