I am trying to learn probability theory through self study, so I have been working through the MIT open courseware problems. I am pretty stuck on the following one enter image description here

How do we find the marginal pdf of $f_Y(y)$? I know we need to integrate the joint over $X$ but I don't see how to do it?

Source: https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-041-probabilistic-systems-analysis-and-applied-probability-fall-2010/assignments/MIT6_041F10_assn05.pdf

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    $\begingroup$ Since this is self study you need the self study tag. Then we can provide hints to help you. $\endgroup$ Feb 11, 2017 at 16:00

1 Answer 1


The following image is the ranges of integration:

$p_Y(y) = \int_x p_{X,Y}(x,y) \, \mathrm{d}x$ $= \int_1^{2-y} \ 3/2 \ dx+\int_y^{1} \ 1/2 \ dx $


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