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MathGroup Archive 2004

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Binomial ratio expectation

  • To: mathgroup at smc.vnet.net
  • Subject: [mg49905] Binomial ratio expectation
  • From: Ismo Horppu <ishorppu at NOSPAMitu.st.jyu.fi>
  • Date: Thu, 5 Aug 2004 09:22:25 -0400 (EDT)
  • Organization: University of Jyvaskyla, Finland
  • Sender: owner-wri-mathgroup at wolfram.com

I have the following problem, I need to compute 
EXPECTATION[X/(2+X)], 
where X follows Binomial distribution with n trials and success 
probability of w.

I have tried to solve it with Mathematica (version 4.1) as
Sum[((x)/(2 + x))*Binomial[n, x]*w^x*(1 - w)^(n - x), {x, 0, n}]

I omit here the result which seems to be okay (according to 
simulations) for values 0<w<1. Problem is that result (intermediate 
or full simplified one) is not defined with values 0 or 1 of parameter w.
However, it is trivial to compute the result by hand on those cases 
(as the X is then a fixed constant, 0 or n). 

Does anyone know how to get the full result with Mathematica, or at 
least a warning that the result is partial. I am also interested in 
whether someone knows what kind of summation formula Mathematica uses 
for the sum, some kind of binomial identity formula perhaps? (I am 
unable to find which one, any references would be appreciated).


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