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I have that $Z = X + 2Y$. $X, Y$ are independent. I know $f_X(x), f_Y(y), f_{X,Y}(x,y)$ and $f_Z(z). $ How can I find $f(x,y|z)$?

I know that $f(x,y|z) = f(x, y, z)/f(z) = f(z| x, y)*f(x, y)/f(z)$ but i'm stuck because I can't find $f(x, y, z)$ or $f(z| x, y)$. Which strategies can I use?

Thanks in advance.

Bruce Kane
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If we assume all continuous variables, $f(z|x,y)=\delta(z-x-2y)$, which means if you know $x,y$ you also know $z$ for certain. In discrete version we just use indicator function instead of dirac delta. A better question would be finding $f(x|z)$ by the way.

gunes
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  • So I can always say that $f(\textbf{x}|z) = f(\textbf{x})/f(z)$ since $f(\textbf{x},z) = f(\textbf{x})$ if $Z$ is a function of $\textbf{x}$, if the variables are continuous, regardless of the function, I assume. Thanks for the answer. – Bruce Kane Jan 08 '19 at 11:16
  • With a small correction. If $g(\textbf{x})=z$ (g is a deterministic function like $x+2y$ above), then $f(\textbf{x}|z) = \delta(g(\textbf{x})-z) f(\textbf{x})/f(z)$. Multiplication of $\delta$ function is essential because it ensures the equation $g(\textbf{x})=z$ always holds. For example, think about $Z=X+2Y$ again, the probability $P(X=x,Y=y|Z=z)$ will be zero when $x+2y=z$ doesn't hold, right? – gunes Jan 08 '19 at 11:30