# Let $X \sim \mathrm{Multinomial} (n, p)$. What is the distribution of $X/n$?

Let $$n, k \in \mathbb N^*$$ such that $$p_1, \ldots, p_k \ge 0$$ and $$p_1 + \cdots + p_k = 1$$. Let $$p := (p_1, \ldots, p_k)$$ and $$X \sim \mathrm{Multinomial} (n, p)$$. This means $$X$$ follows a multinomial distribution consisting of $$n$$ independent trials in which the probability of outcome $$i$$ is $$p_i$$.

Then $$\frac{X}{n}$$ is a vector of probabilities. I would like to ask if the distribution of $$\frac{X}{n}$$ is already investigated and has some simple form.

• Define $Y = X/n$, Then $nY$ has a Multinomial($n,p$) distribution... this may help you find the solution. As a note: $X/n$ isn't a vector of probabilities, it's a vector of observed frequencies. Jul 10, 2022 at 19:01
• Answered for the Poisson distribution at stats.stackexchange.com/questions/240720/…. The answer here will be essentially the same.
– whuber
Jul 10, 2022 at 19:51

The pmf of $$\mathbf{X}$$ is
$$P(\mathbf{X} = \mathbf{x}) = n! \prod_{i=1}^k \frac{p_i^{x_i}}{x_i!}$$
From this you can deduce that the pmf of $$\mathbf{Y} = \mathbf{X}/n$$ is
$$P(\mathbf{Y} = \mathbf{y}) = P(\mathbf{X} = n\mathbf{y}) = n! \prod_{i=1}^k \frac{p_i^{(ny_i)}}{(ny_i)!}$$