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Given the variables $X$ and $Y$ which are correlated and jointly Bivariate with corresponding probability distribution function denoted by $f_{X,Y}(x,y)$, and given the linear relationship

$Z = a+\frac{b}{c}Y,\quad a,b,c\in\mathbb{R}$

How do I get the joint probability distribution $f_{X,Z}(x,z)$.

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    >I got the result by taking the derivative.... BUT that is _not_ what you wrote; you obtained $f_{X,Z}(x,z)$ but that **is not the same** as $P\{X=x,Z=z\}$ which denotes the probability that $X$ has value $x$ and simultaneously $Z$ has value $z$. For continuous random variables, $P\{X=x,Z=z\} = 0$ for all $x$ and $z$.2012-02-13

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One gets $ f_{X,Z}(x,z)=|c/b|\cdot f_{X,Y}(x,(c/b)(z-a)). $ More generally, if $(T,Z)=M\cdot(X,Y)$ and $M$ is invertible, $ f_{T,Z}(t,z)=|\det M|^{-1}\cdot f_{X,Y}(M^{-1}\cdot(t,z)). $

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    Thanks @Dider, so that means what I wrote is right2012-02-16