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Suppose I have the following joint pdf of $X$ and $Y$.

$$f_{X,Y}(x,y) = \begin{cases}c\cdot x \cdot \max(x,y) \text{ if } 0\leq x\leq 1, 0\leq y\leq 1\text{ and }x+y\leq 1;\\ 0 \text{ otherwise}\end{cases}$$

How do I

(a) find the constant $c$,

(b) compute $P(X>2Y)$, and

(c) compute the marginal pdfs of $X$ and $Y$?

Currently, I have the following. For part (a), I calculated this. Is this right?

I apologize for not being able to write in the proper notation on this site. For part (b), I calculated this. Is this right?

What is part (c)? How should I set up the integrals? I am having trouble getting them to equal 1.

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    Hint: Draw a reasonably large sketch (say 10 cm by 10 cm) of the plane with coordinate axes $x, y$ and mark on it the lines $x = y$ and $x+y = 1$ (make the point $(1,0)$ be $5$ cm from the point $(0,0)$). You should be able to see a large right triangle divided into two congruent right triangles in your sketch. $f_{X,Y}(x,y)$ is a surface above the large triangle whose height above the plane at $(x,y)$ is given by two different expressions $cx^2$ and $cxy$ depending on which smaller triangle the point $(x,y)$ lies in. Mark the triangles with the appropriate expressions. _Now_ can you proceed?2012-09-17
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    Yes, I've made significant progress. What do you think?2012-09-17
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    Did you draw the sketch? I have no idea what your WA page is trying to calculate.2012-09-17
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    Yes, there are four distinct regions which are represented by the four double integrals.2012-09-18
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    Got something from my answer?2012-09-30

1 Answers 1

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(a) Integrate the function $f_{X,Y}$ on the set $[0,1]^2$ and choose $c$ such that the result is $1$.
(b) Integrate the function $(x,y)\mapsto f_{X,Y}(x,y)$ on the set $(x,y)\in[0,1]^2$, $x\gt2y$.
(c) Integrate the functions $f_{X,Y}(x,\cdot)$ and $f_{X,Y}(\cdot,y)$.

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    What is the correct form of the integral for part a?2012-09-16
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    Sorry but I do not understand your question.2012-09-16
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    I understand I have to integrate the function and set it equal to one, but what are the boundaries of the two integrals?2012-09-16
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    You seem to want to integrate $cxy$ on $0\lt x\lt y\lt1-x$ and $cx^2$ on $0\lt y\lt x\lt1-y$ (and now, your job is to understand and to check this proposition, maybe it is wrong...).2012-09-16
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    I think it is wrong. What do you think of how I set up the integrals for part (a)?2012-09-17
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    What I think is that I fail to see how the integrals in this WA page are even related to your problem... You might want to read again, slowly, my last comment.2012-09-17