Mathematica 9 is now available
Services & Resources / Wolfram Forums
-----
 /
MathGroup Archive
2002
*January
*February
*March
*April
*May
*June
*July
*August
*September
*October
*November
*December
*Archive Index
*Ask about this page
*Print this page
*Give us feedback
*Sign up for the Wolfram Insider

MathGroup Archive 2002

[Date Index] [Thread Index] [Author Index]

Search the Archive

Re: Re: approximation for partitial binomial sum

  • To: mathgroup at smc.vnet.net
  • Subject: [mg36403] Re: [mg36381] Re: [mg36239] approximation for partitial binomial sum
  • From: Daniel Lichtblau <danl at wolfram.com>
  • Date: Wed, 4 Sep 2002 21:22:17 -0400 (EDT)
  • References: <200209040656.CAA28091@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Constantine wrote:
> 
> 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..
> > [...]
> Office: Taub 411
> Tel: +972 4 8294375

You can get a closed form in terms of special functions if you split
into two cases depending on whether n is even or odd.

In[39]:= n = 2*m;

In[40]:= InputForm[Sum[Binomial[n,k]*p^(n-k)*(1-p)^k, {k,0,m-1}]]
Out[40]//InputForm= 
p^(2*m)*((p^(-1))^(2*m) - ((-4 + 4/p)^m*Gamma[1/2 + m]*
    Hypergeometric2F1[1, -m, 1 + m, (-1 + p)/p])/(Sqrt[Pi]*Gamma[1 +
m]))

In[41]:= n = 2*m+1;

In[42]:= InputForm[Sum[Binomial[n,k]*p^(n-k)*(1-p)^k, {k,0,m-1}]]
Out[42]//InputForm= 
p^(1 + 2*m)*((p^(-1))^(1 + 2*m) - (2^(1 + 2*m)*(-1 + p^(-1))^m*Gamma[3/2
+ m]*
    Hypergeometric2F1[1, -1 - m, 1 + m, (-1 + p)/p])/(Sqrt[Pi]*Gamma[2 +
m]))


Daniel Lichtblau
Wolfram Research


  • Prev by Date: RE: How to define new function using the previous output?
  • Next by Date: Re: 3D plot - resend
  • Previous by thread: Re: approximation for partitial binomial sum
  • Next by thread: Re: approximation for partitial binomial sum