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How can I answer the questions with these information

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  • $\begingroup$ 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)? $\endgroup$ Apr 7, 2021 at 21:59
  • $\begingroup$ Yes it should be f(x,y) $\endgroup$
    – Student
    Apr 7, 2021 at 22:17

1 Answer 1

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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.

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