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MathGroup Archive 2002

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Re: approximation for partitial binomial sum

  • To: mathgroup at smc.vnet.net
  • Subject: [mg36407] Re: approximation for partitial binomial sum
  • From: "Carl K. Woll" <carlw at u.washington.edu>
  • Date: Wed, 4 Sep 2002 21:22:24 -0400 (EDT)
  • Organization: University of Washington
  • References: <al4b46$rgl$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Constantine,

If you break your problem up into two cases, even n and odd n, then
Mathematica can sum up your problem and get results, albeit with
hypergeometric functions. Consider the following (make sure you look at this
with a fixed font):

In[21]:=
evenans = Sum[Binomial[2*n, k]*p^(2*n - k)*(1 - p)^k, {k, 0, n - 1}];
In[22]:=
PowerExpand[FunctionExpand[FullSimplify[evenans, n \[Element] Integers]]]
Out[22]=
     2 n        n  n           1                                  p - 1
    2    (1 - p)  p  Gamma[n + -] Hypergeometric2F1[1, -n, n + 1, -----]
                               2                                    p
1 - --------------------------------------------------------------------
                           Sqrt[Pi] Gamma[n + 1]
In[23]:=
oddans = Sum[Binomial[2*n + 1, k]*p^(2*n + 1 - k)*(1 - p)^k, {k, 0, n - 1}];
In[24]:=
PowerExpand[FunctionExpand[FullSimplify[oddans, n \[Element] Integers]]]
Out[24]=
     2 n + 1        n  n + 1           3
p - 1
    2        (1 - p)  p      Gamma[n + -] Hypergeometric2F1[1, -n - 1, n +
1, -----]
                                       2
p
1 - ------------------------------------------------------------------------
--------
                                 Sqrt[Pi] Gamma[n + 2]


Is this what you were looking for?

Carl Woll
Physics Dept
U of Washington


"Constantine" <celster at cs.technion.ac.il> wrote in message
news:al4b46$rgl$1 at smc.vnet.net...
> Hi.
> I want to get some F and R such that:
>
> F[n,p] + R[n,p] =  Sum[Binomial[n,k] p^(n-k) (1-p)^k, {k, 0, Floor[n/2] -
1}],
> when F[n,p] is an approximation to the sum and the R is the remaining
error.
>
> Thanks in advance for any hint.
> Constantine.
>
> At 06:34 AM 8/28/2002 -0400, you wrote:
>
> >In a message dated 8/28/02 4:44:13 AM, celster at cs.technion.ac.il writes:
> >
> >
> >>I'm looking for a way of finding the approximation for partitial
binomial
> >>sum.
> >>I'll be pleasant for any hint..
> >
> >
> >Use the standard add-on package Statistics`NonlinearFit` to do a
> >NonlinearFit to whatever model you want to use for the approximation.
> >
> >
> >Bob Hanlon
> >Chantilly, VA  USA
>
> Constantine Elster
> Computer Science Dept.
> Technion I.I.T.
> Office: Taub 411
> Tel: +972 4 8294375
>
>




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