# 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.

• What do you call to multiply two distributions?
– Did
Commented Oct 13, 2012 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?) Commented Oct 13, 2012 at 1:04
• Obviously the OP does not intend to answer the two comments above. Exquisite courtesy...
– Did
Commented Oct 13, 2012 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.