Question

Let$r=r1+...+rk,$where all ri are positive integers. Argue that if X1, ... , Xr has a multinomial distribution, then so does Y1, ... , Yk where, with

$r_0=0$,

$Y_i=\sum _{j=r_{i-1}+1}^{r_{i-1}+r_i}{X}_j,i\le k$That is, Y1 is the sum of the first r1 of the Xs, Y2 is the sum of the next r2, and so on

Short answer

If $X_1$and $X_r$have multinomial distributions, then $Y_1$ and $Y_k$ have as well.

Step-by-step solution

To show

If $X_{1}and{X}_r$have multinomial distributions, then $Y_{1}and{Y}_2$also have the same.

Explanation

To given: $Y_i=\sum _{j=r_{i-1}+1}^{r_{i-1}+r_i}{X}_j,i\le k$

To prove: consider

$$\begin{gathered} Y_i=\sum _{j=r_{i-1}+1}^{r_{i-1}+r_i}{X}_j,i\le k \\ r=r_1+........+r_k \end{gathered}$$

When each trial results in one of the outcomes $X_i$indicates the number of each of the types of outcomes $1,\dots ,r$that occur in n independent trials, each with probability $p_1......p_r$. On the other hand, $Y_i$represents a category of outcomes in which the trial resulted in any of the outcome types$1,\dots ,r_1$whereas$Y_2$represents a category of outcomes in which the trial resulted in any of the outcome types $r_1+1........,r_1+r_2$and so on.

However, we can see that $Y_1.......Y_k$has the multinomial distribution by definition.