# Multiple dependent binomial random variables is a multinomial random variable? How to reconcile this with multinomial's requirement for independence?

For $$n$$ independent trials each of which leads to a success for exactly one of $$k$$ categories, with each category having a given fixed success probability, the multinomial distribution gives the probability of any particular combination of numbers of successes for the various categories.

But let's say we have a binomial point process with $$n$$ points over some region of disjoint Borel sets $$B_1, B_2, \dots, B_n$$ (I think "Borel set" is the correct terminology here, but I'm not sure). Let $$Y_i = N_{\mathbf{X}}(B_i)$$ be the random variable describing the number of points in the Borel set $$B_i$$. Then the $$Y_i$$ are themselves binomial random variables, right? But then the $$Y_i$$ are all dependent, since $$Y_1 + Y_2 + \dots, Y_n \le n$$. But all of these dependent binomial random variables together would be a multinomial distribution with parameters $$(n, p_1, p_2, \dots, p_n)$$, right? But how do we reconcile this with the fact that the multinomial distribution requires independence?

I realise that I've used very flimsy language here. I would appreciate it if someone would please tighten-up/correct my probabilistic/mathematical language in their explanation of this.

• The problem here is that you are confusing what process is independent in each case. Note that in the description of multinomial distribution that you are giving, it is each trial that is independent, not the number of successes in each category. These will be marginally binomial, but dependent, just like in the example you described. Aug 16, 2023 at 10:16
• @Julius This seems reasonable – although, it isn't totally clear to me precisely what it means to say that a trial is independent, as opposed to the number of successes. It also isn't clear to me how the canonical definition of independence in probability theory en.wikipedia.org/wiki/Independence_(probability_theory) relates to all of this. Aug 18, 2023 at 8:11
• please see my answer that I've posted. I realise upon a closer reading of your question that part of the confusion might come from the fact that the number of trials and the number of outcomes in a multinomial need not be the same. Aug 18, 2023 at 10:17

For a multinomial distribution with $$n$$ trials and $$k$$ outcomes for each trial, we may sample from each distribution as follows.

We let $$X_1,\dots,X_n$$ represent the outcomes of each of the $$n$$ trials. So each $$X_i$$ takes exactly one of the values $$1,\dots,k$$, and, $$X_i= j$$ with probability $$p_j$$. Importantly, each trial is independent of all others, in the sense that the random variables $$X_1,\dots,X_n$$ are independent.

Now we realise the multinomial distribution as the random vector $$N=(N_1,\dots, N_k)$$ by letting each $$N_j$$ count how many of the $$X_i$$ have value equal to $$j$$, i.e. $$N_j = \sum_{i=1}^n 1_{\{X_i=j\}}.$$

Now, even though the $$X_i$$ (the trials) are independent of each other, note that the $$N_j$$ are not! Indeed, have the condition $$N_1 + \dots N_k = n$$, even though the $$N_i$$ have binomial marginals.

So in short, the condition for independence is for the trials with which we sample the distribution, and not the counts that make up the realisation. You can double check that this agrees with you intuition in the Binomial ($$k=2$$) case, where we have a sequence of (independent) coin flips, and we count the number of heads and tails.

• This is a clarifying answer, but isn't this exactly what you were referring to in the comment, just elaborated upon? Aug 18, 2023 at 12:26
• @ThePointer Yes. I was adding clarification in order to address further your comments, namely about what it means for trials to be independent as opposed to the number of successes. Aug 18, 2023 at 12:29
• Would you please clarify how this relates to the canonical definition of "independence" in probability theory en.wikipedia.org/wiki/Independence_(probability_theory) ? For two random variables en.wikipedia.org/wiki/… , it says that "Two random variables $X$ and $Y$ are independent if and only if (iff) the elements of the $\pi$-system generated by them are independent; that is to say, for every $x$ and $y$, the events $\{ X \le x\}$ and $\{ Y \le y\}$ are independent events ..." Aug 19, 2023 at 6:05
• Or, put another way, how do you reconcile your description with the canonical definition of "independence" in probability theory? Aug 19, 2023 at 8:16
• @ThePointer The requirement for defining the multinomial distribution is that the $X_i$ satisfy the canonical definition of independence. There is no such requirement for the $N_j$. Aug 20, 2023 at 15:10