I found multinomial distribution really powerful in theory. But I don't know if my real case can be treated as multinomial. There are two cases:

  1. There are $n$ students, $30\%$ chose apple, $40\%$ chose banana, $30\%$ chose grape. Then can I say the choice of single student follow a $\mathrm{Multinomial}(n,0.3,0.4,0.3)$ distribution?
  2. Similar, say I have $n$ books, $30\%$ will be distributed to group 1, $40\%$to group 2, the rest to group 3. Then can I assume the allocation of a single book follows a multinomial distribution? What if I was to allocate money instead of books? In this case, money is continuous not discrete.
  • 4
    $\begingroup$ Take care in interpretation. It's conceivable, for example, that the "choice of [any] single student" is not at all random and should not be described by a multinomial distribution. (Some students might hate bananas, for instance.) What a distribution can describe is a process for selecting the $n$ students from a population. For the multinomial distribution to apply, that process has to be analogous to picking slips of paper--one per student--out of a well-mixed bowl of a great many such slips. $\endgroup$
    – whuber
    Jan 8, 2013 at 16:21
  • $\begingroup$ youtube.com/watch?v=PxVnYVScU-o $\endgroup$ Jan 8, 2013 at 16:25

1 Answer 1


When your variable is the choice of fruit, if you extract a sample of n students from a population of N students then Y is distributed as a multinomial. Another example is the extraction of n balls from a urn having red, blue and orange balls, or the extraction of n patients who have been administered a certain treatment (treatmentA, treatmentB and placebo). about the question #2: if it is a feature already present on each book (and mutually exclusive, for example if you want to group by Author) then yes, if you extract a sample of books and observe this feature the distribution of this variable will be distributed as a multinomial.

The same holds for money: if you have a sample of "money" (bank notes and coins) and you extract a sample of n money and observe the feature "kind of money" (bank notes vs coins) you will then have a binomial distribution. You could instead observe another feature, such as the "head" (defined as the name of the person who's drawn on the head side of the coin or on the banknote) of each coin/banknote and that would follow instead a multinomial distribution.

Edit: I am assuming that the N individuals of each population present one and only one category of the characteristic.


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