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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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up vote 2 down vote accepted

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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Maybe this is a too newbie question, how do you know X=x and X+Y =z iff X=x and Y=z-x? My question is focused on "Y=z-x", how that's equivalent to X+Y =z. Thanks. – user1486802 Dec 1 '12 at 3:32
Suppose that $X=5$ and $X+Y=17$. What is $Y$? (Of course I should say suppose $X$ has taken the value $5$, and $X+Y$ has taken the value $17$. But I wanted to simplify the statement.) – André Nicolas Dec 1 '12 at 3:38 I was sorta trying to ask if it was okay to interchange big letter X,Y,Z to small x,y,z. whether it's mathematically/probabilistically valid. Thanks. – user1486802 Dec 1 '12 at 22:51
@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. – André Nicolas Dec 2 '12 at 0:07

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