# Finding the density of the joint distribution $(Y_1,...,Y_m)$ and marginal distribution of $Y_j$

This is a very important problem to me and any assistance would be appreciated and likely to be useful.

let $X_1,...,X_n \sim P(X_i=j) = p_j(j=1,...,m)$ are a total of $n$ independent trials, each of the trials results in exactly one of $m$ possible outcomes, and each outcome has a success probability $0\le p_j\le 1$ $(1 \le j \le m, \sum_{j=1}^{m}{p_j} = 1)$. Define $Y_j = \sum_{i=1}^{n}{I(X_i=j)}$, where $I(X_i = j) =1$ if $X_i = j$ and $0$ otherwise. $Y_j$ is the number of $X$'s equal to $j$ and $\sum_{j=1}^{m}{Y_j} = n$

I'm just trying to find the names of the joint distribution of $(Y_1,...,Y_m)$ and marginal distribution of $Y_j$ so that I can find their expected values, variances, and covariances.

• Each $Y_j$ has a marginal binomial distribution, and they jointly follows a multinomial distribution.
– BGM
Commented Apr 8, 2017 at 12:30

It may be instructive to first think about an easier case when $m = 2$. Then each random variable $X_{i}$ represents an independent coin flip, with probability $p_{1}$ of success (reversing roles of $1$ and $2$ you can think of $p_{2}$ as probability success). It is well known that such scenario is modelled by the Binomial distribution, so that we can calculate:
$$\mathbb{P}(Y_{1} = k, Y_{2} = n-k) = \binom{n}{k}p_{1}^{k}p_{2}^{n-k}$$
In particular, the marginal distribution of $Y_{1}$ is Binomial with parameters $(n, p_{1})$, while the marginal distribution of $Y_{2}$ is Binomial with parameters $(n, p_{2})$.
If $m > 2$, this process is generalized by the Multinomial distribution.