Re: a problem with functions of binomials.


<claudia.giovannoni@xxxxxxxxx> wrote in message
news:1155735244.630717.292720@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
A question with functions of binomial random variables.

Let aj be the generic coefficient of a Binomial with parameters n and
p.

Similarly, let bj be the generic coefficient of a Binomial with
parameters (n+1) and p.

Now, let t be a positive real number and define the function H(y) as
the integral from 0 to y of the CUMULATIVE of a Normal distribution
with mean m and variance s^2.

Note that for any strictly positive t

Sum from 0 to n of ajx(tj/n) = Sum from 0 to n+1 of bjx(tj/n+1)=tp.

Ideally, what I want to show is that for any strictly positive t

Sum from 0 to n of ajxH(tj/n) > Sum from 0 to n+1 of bjxH(tj/n+1)


It is true, and all you need to know about H is that it is strictly convex.
From the identity

j/(n+1) = (j/(n+1)) * ((j-1)/n) + (1 - j/(n+1)) * (j/n)

and the convexity of H,

H(tj/(n+1)) < (j/(n+1)) * H(t(j-1)/n) + (1 - j/(n+1)) * H(tj/n)

You can use this to show your inequality, but I'm not writing it out in
ASCII.

Graham






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