# A binomial multiplied by a poisson

What distribution do you obtain when you multiply a poisson distribution and a binomial distribution, and why? I'm assuming you obtain a poisson distribution.

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What do you call to multiply two distributions? –  Did Oct 13 '12 at 0:41
The problem is underspecified. Are the two distributions independent? (I also don't know what it means to multiply two distributions. Are you taking the product of the corresponding random variables?) –  Qiaochu Yuan Oct 13 '12 at 1:04
Obviously the OP does not intend to answer the two comments above. Exquisite courtesy... –  Did Oct 13 '12 at 8:23

Here is a physical example:

If we interpret $\mu$ and the probability that a photon produces an electron, then for a given number of photons entering the photo detector (say $n$) the probability distribution of electrons coming out is a binomial distribution with n trials and a probability of success of $\mu$.

$$P(m)=\frac{n!}{m!(n-m)!}\mu^m (1-\mu)^{n-m}_{} =\binom{n}{m}\mu^m (1-\mu)^{n-m}_{}$$

The number of photons that go into the binomial distribution is the output of a Poisson distribution. We cannot get more electrons out than photons that went into the photo detector. We can sum up all the possible binomial distributions with a Poisson distribution weighting factor.

$$P(m,\lambda,\mu)=\sum_{j=m}^{\infty} \frac{j!}{(j-m)!m!}\mu^m(1-\mu)^{j-m} \frac{\lambda^j e^{-\lambda}}{j!}$$ Simplify and bring terms that do not depend on j outside of the sum.

$$P(m,\lambda,\mu)= \frac{\mu^m e^{-\lambda}}{m!} \sum_{j=m}^{\infty} \frac{\lambda^j(1-\mu)^{j-m}}{(j-m)!}$$ Let $n=j-m$

$$P(m,\lambda,\mu)= \frac{\mu^m e^{-\lambda}}{m!} \sum_{n=0}^{\infty} \frac{\lambda^{n+m}(1-\mu)^{n}}{n!}$$

$$P(m,\lambda,\mu)= \frac{(\lambda \mu)^m e^{-\lambda}}{m!} \sum_{n=0}^{\infty} \frac{(\lambda (1-\mu))^{n}}{n!}$$

$$P(m,\lambda,\mu)= \frac{(\lambda \mu)^m e^{-\lambda \mu}}{m!}$$

This is a Poisson distribution with a mean of $\lambda \mu$!

Note: the "!" in the last line of text should be understood as excitement and a not a factorial.

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Thank you for the help. –  shmiggens Oct 13 '12 at 1:54