Suppose X = aU + bV, Y = cU + dV. a, b, c, and d are constant. U and V are independent with known pdf, e.g. f(u) and g(v). We know that the pdf of X and Y are the convolution of f and g, but what is the joint distribution of X and Y? I can find the answer for only the case if U and V are normal distributed, but is there any answer if they are not normal?
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Use MathJax to format your questions: https://math.meta.stackexchange.com/q/5020/787310 – Alek Fröhlich Jun 07 '20 at 04:51
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Suppose $X = aU + bV, Y = cU + dV$ where $a$, $b$, $c$, and $d$ are constant. $U$ and $V$ are independent with known pdf, e.g. $f(u)$ and $g(v)$. We know that the pdf of $X$ and $Y$ are the convolution of $f$ and $g$, but what is the joint distribution of $X$ and $Y$? I can find the answer for only the case if $U$ and $V$ are normal distributed, but is there any answer if they are not normal? – Lo Simon Jun 07 '20 at 07:37