Problem with deriving joint/marginal PDF

lolproe

Hi, I'm having a problem with a MATLAB assignment in my probability class. It has to do with finding the joint and marginal PDFs of a pair of random variables. I understand this stuff on paper, but for some reason this assignment is giving me problems since I don't have a good idea of how the random variables translate into a graph.

1. The problem statement, all variables and given/known data
Here are the two questions giving me troubles

The plot produced by test.m is attached as 1.jpg

2. Relevant equations
Nothing as far as I know beyond basic probability stuff and integration

3. The attempt at a solution
My main problem is that I don't conceptually understand how to take a graph of sample data points and translate it into a PDF. After looking at the graph for a while, my guess at the joint PDF is

[tex]f(x,y)=\left\{\begin{array}{cc}1-x,&\mbox{ if }
0\leq x\leq y\leq 1\\0, & \mbox{ otherwise } \end{array}\right.[/tex]

It makes sense to me logically, since for a given y value, x can only be as large as the y value and its probability decreases linearly as x increases from 0 to y. So that should cover part a, but I really don't know if I'm right.

For b, I just integrated it over the range of y, so:

[tex]f(x)=\int_x^1 (1-x)dy = (1-x)y|_x^1 =\left\{\begin{array}{cc}(1-x)^2,&\mbox{ if }
0\leq x\leq 1\\0, & \mbox{ otherwise } \end{array}\right.[/tex]

I'm not really good with the limits during this sort of thing, so I figure this is probably where I made a mistake but I'm not sure.

When I run test.m, it saves X as a variable in MATLAB, and I have also have a function which plots the normalized histogram of a chosen variable with a number of bins. So running this script on X with 10 bins, I get the second attachment, 2.jpg.

This is where things really start confusing me. From looking at that, it seems like the marginal PDF should be linear, or specifically [tex]2(1-x)[/tex] instead of [tex](1-x)^2[/tex]. This would be easy to get from my joint PDF, but I see no way to get my Y limits to be as simple as 0 and 2. This is the main problem I have with the question. This is the first chance I have to really verify my work and it doesn't match up, so I don't know where I made my mistake. Any help would be much appreciated, thanks

[edit]
Looking at the question again, I guess it saying the values are uniformly distributed could lead somewhere but I'm not sure how. I tried assuming the marginal PDF was a uniform distribution with y as the limit but that didn't really get me any closer to the answer.

Attached Thumbnails

   

Ray Vickson

The f(x,y) you wrote above does not integrate to 1, so is not a legitimate probability density function. Instead, try setting f(x,y) = constant inside the triangle.

RGV