2
$\begingroup$

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.

  • 0
    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
  • 0
    Yes, I've made significant progress. What do you think?2012-09-17
  • 0
    Did you draw the sketch? I have no idea what your WA page is trying to calculate.2012-09-17
  • 0
    Yes, there are four distinct regions which are represented by the four double integrals.2012-09-18
  • 0
    Got something from my answer?2012-09-30

1 Answers 1