# Random variables X, G are have functional relationship G=g(X). How does g relate the graphs of their distributions?

More specifically, as an educational tool I want to prepare a slideshow showing (in 2-D) the graph of $F_X$ transforming into the graph of $F_G$. (I think it can be done in 4 steps (e.g., graphs on $R$, then $X$, then $G$, and back up to $R$) and am looking for rigor behind the visuals.)

The thought process:

A random variable, X, is defined as a particular type of function from a probability space to the reals, its image the support of its cdf, FX. Thus, “X” denotes a function and, in a supporting role, a set. Given random variable, G=g(X), transform the graph of FX to that of FG in R2 as follows:

1.{(r,FX(r)): rϵR} to {(x,FX(x)): xϵX}: We simply highlight the sub-graph for X.

2.{(x,FX(x)): xϵX} to {(g(x),FX(x)): xϵX}: For each (x,FX(x)) plot (g(x),FX(x)).

3.{(g(x),FX(x)): xϵX} to {(g,FG(g)): gϵG}: We arrive at the sub-graph for G by “sorting” the (g(x),FX(x)) in place vertically by the size of g(x): the resulting 2nd coordinate in (g,FG(g)) being the “sum” over all the F(x)-F(x-) for those (g(x),FX(x)) for which g(x)≤g .

4.{(g,FG(g)): gϵG} to {(r,FG(r)): rϵR}: As G is the support, if rϵR and not in G then FG(r)=0 or FG(r)=FG(g_r) where g_r is the largest value in G less than r.

At issue in particular is the rigor/validity of Step 3.

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Is $g$ monotonic? – Rahul Jun 14 '12 at 16:57
It can be. However, take as given that g meets the right criteria (i.e., for some sample space S, X:S-->R, g:R-->R is such that composition goX:S-->R is a random variable). – Ace Jun 14 '12 at 19:17

If $g$ is nondecreasing, $F_G(g(x)) = P(g(X) \le g(x)) = P(X \le x) = F_X(x)$.
If $g$ is nonincreasing, $F_G(g(x)) = P(g(X) \le g(x)) = P(X \ge x) = 1 - F_X(x-)$.
It's actually $1-F_X(x-)$, i.e. $1 - \lim_{t \to x-} F_X(t)$. The difference between that and $1 - F_X(x)$ being $P(X=x)$. – Robert Israel Jun 15 '12 at 4:42