Let's say there are two dependent random variables $X$ and $Y$ with joint density function $f$.
What is the PDF of the weighted sum of these two variables, $Z = aX + bY$?
Thanks in advance for any advice or references!
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1$\begingroup$ This is nearly a duplicate of How to add two dependent random variables?, except that you include coefficients $a$ and $b$. I think it might be better to say you are interested in a linear combination rather than just a "sum", since that is dealt with elsewhere on this site. $\endgroup$– SilverfishCommented Feb 27, 2016 at 0:02
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$\begingroup$ (There's a related post here about the sum of discrete random variables.) $\endgroup$– SilverfishCommented Feb 27, 2016 at 0:03
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$\begingroup$ Let $A = aX$ and $B = bY$. Then, the joint density $f_{A,B}(\alpha, \beta)$ is given by $$f_{A,B}(\alpha, \beta) = \frac{1}{|ab|}f_{X,Y}\left(\frac{\alpha}{a},\frac{\beta}{b}\right).$$ Now apply the method described in the question pointed out by Silverfish. $\endgroup$– Dilip SarwateCommented Feb 27, 2016 at 3:28
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