Is this a Multinomial distribution?

Question

I find it hard to notice when do I have a Multinomial distribution and if its possible to "transform" problems into a Multinomial distribution problems.

For example I have the following exercise:

$15$ people come for a test
$21$ quizzes has been printed
$7$ quizzes are type A
$7$ quizzes are type B
$7$ quizzes are type C
Each man get $1$ random quiz

Let X be number of people who got quiz A
Let Y be number of people who got quiz B

Write the joint probability function of X,Y

Now if it would be the joint probability function of X,Y, Z(when Z is Z=number of people who got quiz C) I could easily say its a Multinomial distribution (I think)

But now the sum X and Y is not $1$ (because some people may get C) so how can I answer this question? and how can I know when to search for a Multinomial distribution?

Thanks in advance

Answer 1

Hint

There are 15 people, so you know $Z=15-X-Y$, where $Z$ is the count of people who have test C.   So you have:

$$ \mathsf P(X{=}x, Y{=}y)= \mathsf P(X{=}x, Y{=}y, Z{=}15{-}x{-}y)$$

So, if the sampling is with replacement (different people may select the same puzzle) this is a multinomial distribution.

$$\mathsf P(X{=}x, Y{=}y)_{\text{multinomial}} = \dbinom{15}{x,y,15-x-y} (\tfrac {1}{3})^{15}\quad\big[x\in \{1..7\}, y\in\{1..7\}\big]$$

However, if the sampling is without replacement (everyone gets a distinct quize), this is a multivariate hypergeometric distribution.

$$\mathsf P(X{=}x, Y{=}y)_{\text{hypergeometric}} =\dfrac{\dbinom{7}{x}\dbinom{7}{y}\dbinom{7}{15-x-y}}{\dbinom{21}{15}}\quad\big[x\in \{1..7\}, y\in\{8{-}x..7\}\big]$$