Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
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
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.

share|cite|improve this answer
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.