Q.6.4 - A First Course in Probability

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 

Step-by-Step Solution

Answer

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

Step 1: To show

If $X_1 and X_r$have multinomial distributions, then  $Y_1 and Y_2$also have the same.

Step 2: 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.