[R] Sum of Bernoullis with varying probabilities

[(Ted Harding)]

Many thanks for your comments, Deepayan; and I liked your
recursive solution! Fun indeed.

Just a comment (below) on one of your comments (the rest
snipped).

On 06-Oct-06 Deepayan Sarkar wrote:
> On 10/6/06, Ted Harding <Ted.Harding at nessie.mcc.ac.uk> wrote:
>> Hi Folks,
>>
>> Given a series of n independent Bernoulli trials with
>> outcomes Yi (i=1...n) and Prob[Yi = 1] = Pi, I want
>>
>>   P = Prob[sum(Yi) = r]  (r = 0,1,...,n)
>>
>> I can certainly find a way to do it:
>>
>> Let p be the vector c(P1,P2,...,Pn).
>> The cases r=0 and r=n are trivial (and also are exceptions
>> for the following routine).
>>
>> For a given value of r in (1:(n-1)),
>>
>>   library(combinat)
>>   Set <- (1:n)
>>   Subsets <- combn(Set,r)
>>   P <- 0
>>   for(i in (1:dim(Subsets)[2])){
>>     ix <- numeric(n)
>>     ix[Subsets[,i]] <- 1
>>     P <- P + prod((p^ix) * ((1-p)^(1-ix)))
>>   }
> 
> Don't you want to multiply this with factorial(r) * factorial(n-r)? I
> assume combn() gives all combinations and not all permutations, in
> which case you are only counting
> 
> prod((p^ix) * ((1-p)^(1-ix)))
> 
> once instead of all the different ways in which ix could be permuted
> without changing it.

You had me worried for a moment -- the interplay between perms
and combs is always a head-spinner -- but since I'd verified
already that I get sum(P)=1 when I do this over all values of r,
I had to conclude that your comment must be incorrect.

In fact, if you consider the event "r out of the n gave Y=1",
this can be decomposed into disjoint sub-events

  subset {1,2,3,...,r} gave Y=1, the rest 0
  subset {1,2,3,...,(r-1),(r=1)} gave Y=1, the rest 0.
  ....

and so on over all choose(n,r) subsets, and the probability of
"r out of n" is the sum of the probabilities of the sub-events.
For any one of these (say the first above), the probability is

  P(Y1=1 & Y2=1 & ... & Yr=1 & Y[r+1]=0 & ... & Y[n]=0)

which is simply the product of their probabilities. No further
permutation is involved.

I hope I've got this right (we'll soon find out if not)!

Best wishes,
Ted.

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Date: 06-Oct-06                                       Time: 22:15:44
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