What is the continuous distribution version of multinomial distribution?

Question

I am trying to model a distribution, on the number of occurrences of an event in a 24 hour time span.

Right now, I discretize the 24 hour time span into hourly intervals, and each hour is taken as a categorical outcome, and I count the number of occurrences of the event in each hour (outcome). Hence, this problem is modeled as a multinomial distribution.

As time is a continuous variable, is there a continuous version of multinomial distribution? That is, I can count the number of occurrence of the event in the continuous outcome(time).

Why a multinomial distribution? Is there a fixed total number of occurances? The logical model to use would be a Poisson process. Anyway, the continuous analogue of the multinomial distribution is the multivariate normal distribution. — Raskolnikov, Mar 02 '12 at 22:32

score 3 · Accepted Answer · edited Jan 28 '15 at 02:27

3

A similar distribution would be the Dirichlet distribution.

A random sample of a Dirichlet distribution is a set of probabilities that add to one. You can then multiply each by, say, $24$, to get a "continuous multinomial distribution."
This is just a direct answer to your question about "continuous multinomial distribution", whether you should use it to model your data is another question.

edited Jan 28 '15 at 02:27

daOnlyBG

2,727

answered Jan 28 '15 at 01:48

qkhhly

234

It doesn't seem at all clear that this addresses the question. You still have these 24 discrete bins into which you're placing each "occurrence", thus you still treat time as discrete. – Michael Hardy Jan 11 '20 at 17:51
I've down-voted this because I think it misconstrues the question, as explained in my comment above conjoined with my posted answer. – Michael Hardy Jan 11 '20 at 18:01
Upvoting because it’s actually useful – william_grisaitis Jun 02 '21 at 17:22

score -1 · Answer 2 · answered Jan 11 '20 at 18:00

It is not clear that the answer from "daOnlyBG" addresses the question at all. On the other hand, the question itself is less than fully clear.

The multinomial distribution, used as described in the question, is appropriate ONLY if the total number of occurrences is NOT RANDOM, so the only thing that is random is which of the one-hour periods each "occurrence" falls into.

A continuous analog is the probability distribution of the order statistics from a continuous uniform distribution on the 24-hour period.

Thus we have $T_1,\ldots,T_n\sim\text{i.i.d.}\operatorname{Uniform} \left[0\text{ hours}, 24\text{ hours} \vphantom{\tfrac22}\right]$ and then we sort them into increasing order, getting $S_1<S_2<S_3<\cdots< S_n$ as the times of the occurrences.

Here the number of occurrences, $n,$ is NOT RANDOM, but the times when they occur are random.

The probability distirbution of $(S_1,\ldots,S_n)$ has a simple formula, but I don't know a standard name for it.

What is the continuous distribution version of multinomial distribution?

2 Answers2