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As you can see from the image below, how does Px,z(x,z) change to Px,y(z-x)? Can someone tell me the logic behind it or, refer me to a link that talks about the similar topic? Thanks alot!!

enter image description here

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Note that $Z=X+Y$. So we have $(X,Z)=(x,z)$ if and only if $(X,X+Y)=(x,z)$. But $X=x$ and $X+Y=z$ iff $X=x$ and $Y=z-x$.

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    @shle2821: In principle, one can use any kind of letter for anything. In practice, it is **very** useful to use caps for random variables, and, as much as practical, to use the accompanying small letter for a *value* the random variable takes. That helps to keep in front of your eyes the important distinction.2012-12-02