How do you sample from a multinomial distribution?
What is the probability mass function for a multinomial distribution?
The multinomial distribution is a joint probability distribution that is sampled whenever you do $n$ trials in which each trial can have one of $k$ possible, mutually-exclusive outcomes. The $k$ random variables that are generated when this distribution is sampled count the number of trials that were equal to each outcome. This definition is explained in the video below:
https://www.youtube.com/watch?v=T88yLmrYYZg&t=411s
The joint probability mass function for the multinomial distribution is given by:
<aside> 💡 $P(X_1=x_1 \wedge X_2 = x_2 \wedge \dots \wedge X_k = x_k ) = \frac{n!}{\prod_{i=1}^k x_i!} \prod_{i=1}^k p_i^{x_i}$
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$p_i$ here is the probability of observing outcome $i$ in a single trial and you thus need to have:
$\sum_{i=1}^k p_i =1$
n, meanwhile, is the total number of trials performed and must therefore satisfy:
$n = \sum_{i=1}^k x_i$
<aside> 📌 SUMMARY: To sample from the multinomial distribution you perform $n$ trials that have $k$ possible outcomes and count the number of times each of the various outcomes appears.
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