MaplePrimes - answers and comments on Question, Integration for finding k-th probability

hm ...

Axel Vogt — Mon, 02 Jul 2012 15:18:06 Z

I do not have Maple at hand right now, so can not check on input errors
(which may occur using the standrad interface)

But in the displayed code you set x:=0, while it also is a variable for integration.

That would explain the shown error:

you want x = 0 ... t but the system gets 0 = 0 ... t

1/7*(1 - (2/3)^(k+1))

Axel Vogt — Wed, 04 Jul 2012 23:05:01 Z

Your function c(k) writes without almost any modification as 

  F := k -> 
    Int(Int(2*(-3*x+7*t)^k/GAMMA(k+1)*exp(x-7*t),x = 0 .. t),t = 0 .. infinity); 

                 infinity    t
                /           /                k
               |           |   2 (-3 x + 7 t)  exp(x - 7 t)
    F := k ->  |           |   ---------------------------- dx dt
               |           |           GAMMA(k + 1)
              /           /
                0           0

This is done by switching to uppercase Int (= 'inert') and combining exponentials

  subs(int = Int, c(k)); combine(%, exp); F:=unapply(%, k);

Instead of using the somewhat more simple version H(k) already given (missing any
reply, test results or confirmation - so I guess statements are enough for you)
finally I arrived at a complete solution. At least I think so. And just check on
values of those k, for which you are able to compute, i.e. c(k) = F(k) = f(k) = H(k).

  f:= k -> 1/7*(1 - (2/3)^(k+1));

But then summing that over k can not converge.

Edited: the sheet attached MP_2dim_integral_erl.mws

seems to be the reason:

Axel Vogt — Mon, 02 Jul 2012 22:31:17 Z

I get c(k) = 95/729, omitting x:=0.

Where I would use UpperCase Int in the definitions and finally would call 'value'
to get the result.

Sum(c(k), k = 0 .. 20); value(%);

  198750905161/73222472421

sum upto infinity

dj_gssst — Tue, 03 Jul 2012 16:24:08 Z

Dear experts,

Thank you for your prompt reply and help. I am able to compute the sum upto 100. But unable to compute it for infinity althought the expression is converging.

one step ahead, at least

Axel Vogt — Tue, 03 Jul 2012 22:59:58 Z

I think your c(k) writes as

H := k -> 2*3^k/GAMMA(k+1) * 
  Int(exp(-14/3*t)*(-GAMMA(k+1,7/3*t)+GAMMA(k+1,4/3*t)),t = 0 .. infinity)

I currently miss familiarity with the Gamma distribution to simplify more.