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Problem with deriving joint/marginal PDF 
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#1
Apr1212, 05:27 PM

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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 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}1x,&\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 (1x)dy = (1x)y_x^1 =\left\{\begin{array}{cc}(1x)^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(1x)[/tex] instead of [tex](1x)^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. 


#2
Apr1212, 11:31 PM

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