# Binomial within a multinomial distribution

Let $\textbf{X} = (X_1, . . . ,X_k)$ denote the random vector of counts, and let $\textbf{x} = (x_1, . . . , x_k)$ denote a possible value for that random vector. Finally, let $f (x | n, \textbf{p})$ denote the joint p.f. of $X$. With $\textbf{p} = (p_1, \dots , p_k)$. There $f$ denotes the multinomial distribution.

We have to prove that $Y = X_1 + X_2 + \dots + X_l$ where $l < k$. So we have to prove that $Y$ is distributed as a binomial distribution with parameters $n$ and $\sum_{i=1}^{l}p_i$.

Here is my approach and I am stuck at a point.

Let $X_i = X_{i_1} + X_{i_2} + \dots + X_{i_n}$

Where $X_{i_1}$ is $1$ with probability $p_i$ or $0$ otherwise. $X_{i_1}$ denotes whether we get object $i$ in first place, similarly for others.

Now $$Y = X_{1_1} + X_{1_2} + \dots + X_{1_n} \\ + X_{2_1} + X_{2_2} + \dots + X_{2_n} \\ \vdots \\ + X_{l_1} + X_{l_2} + \dots + X_{l_n}$$

Now grouping in this way

$$Y = X_{1_1} + X_{2_1} + \dots + X_{l_1} \\ + X_{1_2} + X_{2_2} + \dots + X_{l_2} \\ \vdots \\ + X_{1_n} + X_{2_n} + \dots + X_{l_n}$$

Now let us discuss for $X_{1_1} + X_{2_1} + \dots + X_{l_1}$, this random variable can take a value $1$ or $0$. It is $1$ when any one of it happens ie first slot is a object from $1$ to $l$ which happens with a proability of $\sum_{i=1}^{l}p_i$. Similarly for the other rows. Now each row is a Bernoulli with probability $\sum_{i=1}^{l}p_i$ and there are $n$ rows. But are these Bernoulli variables independent. Because if they were independent I could conclude that this is a Binomial with the required parameters.