MaplePrimes - answers and comments on Question, Statistics: How to create parametric distribution

One way

Markiyan Hirnyk — Tue, 12 Jul 2011 08:47:41 Z

For example,

PS. Your formula for PDF is not correct: look in Wiki.

General approach

Markiyan Hirnyk — Wed, 13 Jul 2011 18:02:24 Z

This can be done in such a way:

> restart; with(Statistics):
> a := (2/mu*(1/2))*sqrt(2)*exp(-(1/2)*(t-sigma)^2/mu^2)*(1/2+(1/2)*erf((1/2)*alpha*(t-sigma)*sqrt(2)/mu))/(sqrt(Pi)*mu);
> S := proc (x, y, z) Distribution(PDF = unapply(eval(a, [sigma = x, mu = y, alpha = z]), t)) end proc;
> SkewNormal(x,y,z):=RandomVariable(S(x,y,z)):
> PDF(SkewNormal(2, 1, -1), t);
> CDF(SkewNormal(2, 1, -1), t);
> Mean(SkewNormal(2, 1, -1), numeric);
1.435810416
> Variance(SkewNormal(2, 1, -1), numeric);

0.6816901

Yes, this works, but how can I manipulate mu and sigma

lu4 — Tue, 12 Jul 2011 11:57:32 Z

Hello Markiyan! Your solution works, but how do you manipulate mu and sigma afterwards? What should I do when I will want to set mu or sigma?

For example for LogNormal distribution I would do the following:

DensityPlot(LogNormal(-.7098242994, 1.049302140))

By eval

Markiyan Hirnyk — Tue, 12 Jul 2011 12:24:18 Z

@lu4 It can be done by the eval command:

> eval(PDF(SkewNormal, x), [mu = 1, sigma = 2, alpha = -1]);

PS. Another way is as follows.

and so on.

@Markiyan Hirnyk Thanks, I've used

lu4 — Wed, 13 Jul 2011 00:40:06 Z

@Markiyan Hirnyk

Thanks, I've used the following code:

SkewedNormal := (xi, omega, alpha) -> Distribution(PDF = ((x) -> x*sqrt(2)*exp(-(1/2)*(x-xi)^2/omega^2)*(1/2+(1/2)*erf((1/2)*alpha*(x-xi)*sqrt(2)/omega))/(omega*sqrt(Pi))), CDF = ((x) -> 1/2+(1/2)*erf((1/2)*(x-xi)*sqrt(2)/omega)-(int(exp(-(1/2)*(x-xi)^2*(1+x^2)/omega^2)/(1+x^2), x = 0 .. alpha))/Pi), Mean = xi+omega*alpha*sqrt(2/Pi)/sqrt(1+alpha^2), Variance = omega^2*(1-2*alpha^2/(sqrt(1+alpha^2)^2*Pi)), MGF = ((x) -> 2*exp(xi*x+(1/2)*omega^2*x^2)*(1/2+(1/2)*erf((1/2)*omega*alpha*x*sqrt(2)/sqrt(1+alpha^2)))))

Somewhat by hand

Markiyan Hirnyk — Wed, 13 Jul 2011 08:36:32 Z

@lu4 Such approach is better than the usage of the eval command. It is somewhat by hand as Maple 13 (I think Maple 15 too) does not find the integrals corresponding to the mean and the variance of this distribution.

Regard, MH

PS. Also look at

Download scewed.mw