# Multinomial Distribution Proof.

Suppose that there are $k+ 1$ different results of a random experiment and that $p_i$ is the probability of obtaining the $i$th result for $i = 1, · · · , k + 1$ (note that $\sum _{i=1}^{k+1} p_i$) Let $X_i$ be the number of times the $i$th result is obtained after $n$ independent repetitions of the experiment. Verify that the joint density function of the random variables $X_1,..., X_{k+1}$ is given by:

$$f(x_1,..., x_{k+1})= \frac{n!}{\prod_{i=1}^{k+1} x_i!} \prod_{i=1}^{k+1} p_i^{x_i}$$

where $x_i= 0,..., n$ and $\sum _{i=1}^{k+1}x_i= n$.

I don't know where to start from to prove that statement, the only clear thing that I have is that the multinomial distribution is given as:

$$Mult(n, p_1, p_2, p_3, ..., p_n) = {n \choose x_1, x_2, ..., x_n} p_1^{x_1} ...p_n^{x_n}.$$

• It is not clear what your question is. How is the multinomial distribution derived? Or how from the formula of multinomial distribution the given probability mass function turns out? – NCh Apr 13 '17 at 1:09