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I'm trying to solve this problem in my homework assignment and I get different result from the answer. I know the answer is right, but at the same time I also don't see where I did wrong in my solution. So here's the problem:

Let X and Y have a joint uniform distribution on the triangle with vertices (0,0), (3,0), (0,3). Find: (i) E(X|Y) and E(Y|X) (ii) Var(X|Y) and Var(Y|X) (iii) EX and Var(X)

I've correctly completed the first and second sub-problem, but for the third one, I struggle to get it right. I'm trying to use the fact that EX = E(E(X|Y)) to derive EX. And the way I approach it is: EX = $$ \int {E(X|Y)*f(Y)} dY$$ where f(Y) is the pdf of Y being a certain value From the sub-problem 1, I have: E(X|Y) = (3-Y)/2 and f(Y) = 3-Y Hence, I tried to solve integration: $$\int{\frac{(3-Y)^2}{2}} dY $$ where 0<=Y<=3 and I get result 4.5 whereas the correct answer is 1.

Can someone please show where I did wrong? I'd much appreciate your help!

Thanks in advance!

Vol_Smile
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1 Answers1

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The joint density is $\frac{1}{4.5}$ over the triangle. That has not been used in calculating the pdf of $Y$, which is $\frac{3-y}{4.5}$ on $[0,3]$ and $0$ elsewhere.

André Nicolas
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  • how come I didn't see this?! I kept using 1 as the joint distribution. Thanks a lot for pointing it out! – Vol_Smile Oct 06 '13 at 07:11
  • Because quite properly you were concentrating on the ideas. – André Nicolas Oct 06 '13 at 07:15
  • I'm stuck again, on the part Var(X). I get result 1/2 where as the answer id 1/6. I used VAR(X) = E(X^2) - [E(X)]^2 in this case. Since I already know E(X)=1, I only need to solve E(X^2), which is equivalent to E(E(X^2|Y)).

    From previous step, I get E(X^2|Y)=(3-Y)^2 / 3 and since pdf of Y is (3-Y)/4.5, I can use both of them and solve the integral to get 3/2 as E(X^2). But that leads to Var(X) = 3/2 - 1 = 1/2

     – Vol_Smile Oct 06 '13 at 08:00
  • The (unconditional) expectation of $X^2$ is indeed $\frac{3}{2}$. Since I am error prone, I checked it two ways. The variance is not $\frac{1}{6}$. – André Nicolas Oct 06 '13 at 17:07
  • Thanks a lot for confirming my result:) I'll let my professor and class know. – Vol_Smile Oct 07 '13 at 04:10