# Is there something called a "Poisson Binomial Distribution"?

• Dec 20th 2010, 05:33 PM
ouiouiwewe
Is there something called a "Poisson Binomial Distribution"?
I've come across this distribution as I was searching for something related to the sum of Binomial variables. The pdf, expected, and variance seem to make sense. However, I have not seen any textbooks using this distribution nor the name being used anything outside of Wikipedia.

Is this distribution known under a different name?

Wikipedia Article:
Poisson binomial distribution - Wikipedia, the free encyclopedia

Original Publication:
http://www3.stat.sinica.edu.tw/stati...dpdf/A3n23.pdf
• Dec 23rd 2010, 09:31 PM
theodds
Well, it makes sense.

From a statistical point of view, without restrictions on the values that the p_i take, it would be very hard to conduct any kind of meaningful inference about the p_i if they were unknown.

For what it's worth, it seems clear that a sum of binomial random variables produces one of these guys (a Poisson-Binomial, that is), with some restrictions on the values the p_i can take. In the special case where the original binomial random variables all have the same success probability, you just end up with another binomial, which is exactly what you would expect.

For example, if \$\displaystyle X\$ is distributed \$\displaystyle bin(n, p_1)\$ and \$\displaystyle Y\$ is distributed \$\displaystyle bin(m, p_2)\$ then \$\displaystyle X + Y\$ will be distributed as a Poisson-Binomial with parameters \$\displaystyle (\underbrace{p_1, ..., p_1}_n, \underbrace{p_2, ..., p_2}_m)\$.
• Dec 23rd 2010, 09:44 PM
ouiouiwewe
Quote:

Originally Posted by theodds
Well, it makes sense.

From a statistical point of view, without restrictions on the values that the p_i take, it would be very hard to conduct any kind of meaningful inference about the p_i if they were unknown.

For what it's worth, it seems clear that a sum of binomial random variables produces one of these guys (a Poisson-Binomial, that is), with some restrictions on the values the p_i can take. In the special case where the original binomial random variables all have the same success probability, you just end up with another binomial, which is exactly what you would expect.

For example, if \$\displaystyle X\$ is distributed \$\displaystyle bin(n, p_1)\$ and \$\displaystyle Y\$ is distributed \$\displaystyle bin(m, p_2)\$ then \$\displaystyle X + Y\$ will be distributed as a Poisson-Binomial with parameters \$\displaystyle (\underbrace{p_1, ..., p_1}_n, \underbrace{p_2, ..., p_2}_m)\$.

Yes, exactly. However, I have not seen this distribution before. Even though I am quite certain it is correct, I still want to know if it is a distribution accepted by the statistics community. If I don't see evidence that it is, then that means I'd have to read the paper myself and see if the proof makes sense.

By the way, it is cool that this board supports TeX.
• Dec 24th 2010, 07:59 AM
theodds
I don't think I understand why you are ambivalent. Whether it is or is not a distribution doesn't depend on whether or not most statisticians know about it or find it useful.

At any rate, the distribution is defined to be the distribution of the sum of a bunch of Bernoulli random variables. There's nothing to prove in that respect. They obviously have the mgf/chf correct. Maybe you would be interested in the proof that the pmf is what they say it is, but from just glancing it looks like exactly what you would expect it to be.
• Dec 24th 2010, 11:56 AM
ouiouiwewe
Yes, I checked the pdf already. It seems correct. Still, I am rather curious as to why this distribution is not included in the textbooks. It's seems to be a very fundamental and important distribution with the Binomial distribution being a special case of it.
• Dec 24th 2010, 03:22 PM
theodds
Quote:

Originally Posted by ouiouiwewe
Yes, I checked the pdf already. It seems correct. Still, I am rather curious as to why this distribution is not included in the textbooks. It's seems to be a very fundamental and important distribution with the Binomial distribution being a special case of it.

Probably because it isn't used very much in practice. I don't see why you would typically want to add binomials together. You can concoct situations, sure, but it probably just doesn't come up very much.
• Dec 29th 2010, 08:39 AM
HHsts
Hi,

I think I have seen a book/text which talks about the addition of binomially distributed events. I will try to find the books name out. As I am on holidays so far this will take by next week.

The text book talked about this in association with the CONVOLUTION which allows merging two PDFs of events which happen independently.

For example, if \$\displaystyle X\$ is distributed \$\displaystyle bin(n, p_1)\$ and \$\displaystyle Y\$ is distributed \$\displaystyle bin(m, p_2)\$ then \$\displaystyle X + Y\$ will be distributed as a Poisson-Binomial with parameters \$\displaystyle (\underbrace{p_1, ..., p_1}_n, \underbrace{p_2, ..., p_2}_m)\$.[/QUOTE]