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
>
>