X has uniform (0,1) and Z has uniform (-1,0). Let Y = X+Z where X and Z are independent. Would Y just be uniform (-1,1)? If so, how would you find the joint pdf of X and Y?
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No. $X+Z$ is less likely to take small or large values. This is similar to how the sum of two rolled ($6$-sided) dice is much less likely to be $2$ than $7$.
You can get the pdf by $$f_Y(x)= \int_0^1 f_X(y) f_Z(x-y) \, \mathrm{d}y,$$ where $f_X$ and $f_Z$ are the pdfs of $X$ and $Z$. See this question: link
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