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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!

Silverfish
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Abc123
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    This is *nearly* a duplicate of [How to add two dependent random variables?](http://stats.stackexchange.com/q/21549/22228), 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. – Silverfish Feb 27 '16 at 00:02
  • (There's a [related post here about the sum of discrete random variables](http://stats.stackexchange.com/q/115198/22228).) – Silverfish Feb 27 '16 at 00:03
  • 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. – Dilip Sarwate Feb 27 '16 at 03:28

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