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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
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1 Answer 1

up vote 1 down vote accepted

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, $F_G(g(x)) = P(g(X) \le g(x)) = P(X \ge x) = 1 - F_X(x-)$.

Other cases are more complicated.

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Is that 1-F(x) in the nonincreasing case? –  Ace Jun 14 '12 at 21:53
    
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
    
"Reading into" the sequence of equalities (and to cover the g constant case) led to something like: (1.) nondecreasing; FG(g)=FX(x) with x from limsup{g(x)=g} (2.) nonincreasing; FG(g)=1-FX(x-), with x from liminf{g(x)=g}. These two cases actually cover most of what I need "for work"(e.g, split up X according to where g is going up, down or constant). Helps with the sorting idea in Step 3. –  Ace Jun 16 '12 at 16:45
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