# What would be the most potent free CAS for mathematical statistics?

I am looking for a free CAS (Computer Algebra System) that is most able to do mathematical statistics. I thought of SAGE, but it doesn't seem to be very powerful.

It can't derive t-distribution : http://ask.sagemath.org/question/1485/student-t-pdf

And it seems to have trouble calculating even the mean and the variance of Laplace distribution!

Here's sage input/output.

INPUT:

def f1(x,mu,sigma):
return 1/sqrt(2*pi*sigma^2)*exp(-1/2*((x-mu)/sigma)^2)
def f2(x,mu,sigma):
return 1/sqrt(2*sigma^2)*exp(-sqrt(2)*abs((x-mu)/sigma))

var("mu sigma")

integral(x*f2(x,mu,sigma), x, -oo,oo)


OUTPUT:

1/2*sqrt(2)*integrate(x*e^(-sqrt(2)*abs(-mu + x)/sigma), x, -Infinity,
+Infinity)/sigma


INPUT:

integral(x*f1(x,mu,sigma), x, -oo,oo)


OUTPUT:

mu


INPUT:

integral((x-mu)^2*f2(x,mu,sigma), x, -oo,oo)


OUTPUT:

1/2*sqrt(2)*integrate((mu - x)^2*e^(-sqrt(2)*abs(-mu + x)/sigma), x,
-Infinity, +Infinity)/sigma


INPUT:

integral((x-mu)^2*f1(x,mu,sigma), x, -oo,oo)


OUTPUT:

sigma^2


Does anyone know free CAS which can calculate t-distribution and the mean and the variance of Laplace distribution with parameters unkown?

-
Insert assume(sigma>0) before computing the integral. Tried it on www.sagenb.com, and it works. Edit: also, it'll be hard to find a more complete free CAS than Maxima (which is what Sage uses). –  mogron Jul 10 '12 at 9:17
Wow, it really works! thank you. Actually I used version 4.5.2. I think it's improved. Anyway, I somehow found a way out, which is using "integral((x-mu)^2*f2(x,mu,sigma), x, mu, oo)+integral((x-mu)^2*f2(x,mu,sigma), x, -oo, mu) ", "factor(integral((mu-x)*f1(x,mu,sigma), x, -oo,mu)+integral((x-mu)*f1(x,mu,sigma), x, mu,oo)) ", and "integral(abs(x-mu)*f2(x,mu,sigma), x, mu, oo)+integral(abs(x-mu)*f2(x,mu,sigma), x, -oo, mu) ". Obviously, it is no more needed. –  KH Kim Jul 10 '12 at 11:40