By Gamma distribution

Markiyan Hirnyk — Fri, 15 Mar 2013 11:56:11 Z

Up to http://en.wikipedia.org/wiki/Moment_generating_function ,
this can be done as follows.
1. MGF := t -> 1/(1-3*t)); #We will not use the condition t < 1/3.
E(X) = eval(diff(MGF(t), t), t = 0);
    3
2. Variance(X)= E(X^2)-E(X)^2 = eval(diff(MGF(t), t,t), t = 0) - 3^2;
   9
3. The function piecewise(t<1/3,1/(1-3*t),undefined) is the MGF of the random variable
Gamma(b, c) with b=3, c=1 (for example, see http://www.maplesoft.com/support/help/Maple/view.aspx?path=Statistics/Distributions/Gamma and
http://en.wikipedia.org/wiki/Gamma_distribution for info).
Let us find its CDF
> with(Statistics); X := RandomVariable(Gamma(3, 1));
                               _R
> CDF(X, t);

        piecewise(t <= 0, 0, 0 < t, -exp(-(1/3)*t)+1)

and the probability P(4.1 < X <4.7)

CDF(X, 4.7)-CDF(X, 4.1);
0.0462155726364304
It should be noticed that the Gil-Pelaez formula http://en.wikipedia.org/wiki/Characteristic_function_%28probability_theory%29 fails in Maple
because of slow convergence:

>Digits := 10: F[X](4.1) := evalf(1/2-(int(evalc(Im(exp(-(I*4.1)*t)*1/(1-3*exp(-I*t))/t,
t = 0 .. infinity, numeric, method = _d01amc))/Pi);
Error, (in int) NE_QUAD_NO_CONV:
The integral is probably divergent or slowly convergent.

Edit. Bad copy and paste.