Say we have $T$ independent Multinomial random variables $X_1,X_2\dots X_T$, with $p(X_t=m)=p_{t,m},m\in\{1,2,...M\}$. What would be the distribution of $X_1+X_2+...+X_T$? If there is no closed-form, is there any approximation version of this distribution? Like, is there a multivariate version of possion-binomial distribution?

  • 1
    $\begingroup$ Could you explain the sense in which $X_1+\cdots + X_N$ could be considered "multivariate"? One wonders whether you use the term "Multinomial random variables" in the standard sense of, say, Wikipedia. $\endgroup$ – whuber Sep 20 '19 at 18:46
  • $\begingroup$ Your question is ambiguous/unclear. If you're summing the counts in a subset of the categories for a single multinomial, the sum will be binomial. $\endgroup$ – Glen_b -Reinstate Monica Sep 22 '19 at 4:01

Assuming you mean multinomial distribution in the usual sense (in which case you should correct the range to include zero.) Then: If the probability parameter $p=(p_1, \dotsc, p_k)$ are all equal, then the sum is also multinomial. If they are unequal, then we have the situation that in the binomial $k=2$ case leads to Poisson-Binomial distribution, and in the multinomial case is known as (surprise!) Poisson-Multinomial. Here is a relevant math SE post.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.