In article <x5rf6.393$[EMAIL PROTECTED]>,
Alan Miller <amiller @ vic.bigpond.net.au> wrote:
>Kumara Sastry wrote in message <[EMAIL PROTECTED]>...
>>Suppose, X ~ Binomial(n1,p1), Y~Binomial(n2,p2) , and X and Y are
>>independent. Also, Z = X+Y. Can anyone please comment on what the pdf
>>of Z is?
>>Thanks
>>Kumara
>Pr(Z=z) = Sum from l to u of p1(r).p2(z-r)
>where p1 is the first binomial probability, and p2 is the second.
>The upper & lower limits of summation, l & u, are not necessarily
>0 and z but:
>l = max(0, z-n2) and u = min(z, n1)
>I hope I have got that right.
>I doubt if the sum simplifies much.
I do not know of any method to simplify the sum.
However, if n1+n2 is sufficiently large, and neither
n1 nor n2 is small, the Fast Fourier Transform can
be used; this even applies to arbitrary convolutions,
but is simplified here by the simplicity of the
characteristic function. Be sure to use at least
2(n1+n2+1) for the vector length for the discrete
Fourier transforms and the inversion. Also, watch
our for roundoff error; if good relative accuracy
is wanted in the tail probabilities, it is not that
likely to be available by transform methods.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
[EMAIL PROTECTED] Phone: (765)494-6054 FAX: (765)494-0558
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