Q.6.60 - A First Course in Probability

Question

In Example $8$b, let $Y_{k+1}=n+1-\sum _{i=1}^k{Y}_i$ Show that $Y_1,....,Y_k,Y_{k+1}$ are exchangeable. Note that $Y_{k+1}$ is the number of balls one must observe to obtain a special ball if one considers the balls in their reverse order of withdrawal. 

Step-by-Step Solution

Answer

As a result, the joint mass function is symmetric, proving the conclusion. That is to say, the random variables $Y_1,....Y_k,Y_{k+1}$ are exchangeable.

Step 1 : Joint probability mass function :

The joint probability mass function is a function that fully describes a discrete random vector's distribution.

Step 2 : Explanation :

The number of balls that must be observed in order to achieve a special ball is $Y_{k+1}$.

$Y_{k+1}=n+1-\sum _{i=1}^k{Y}_i$

$Y_{k+1}$ stands for the amount of balls that must be observed in order to achieve a special ball.

Let $Y_2$ be the number of further balls that are removed until the second special ball arrives, and let $Y_1$ signify the selection number of the first special ball that was withdrawn.

After that, the joint mass function,

$$\begin{gathered} P(Y_j=i_{j,}j=1,2,....,k+1) \\ =P(Y_j=i_{j,}j=1,2,....,k)P(Y_{k+1}=i_{k+1}:Y_j=i_j,j=1,2,....k) \\ =\frac{k!\left(n-k\right)!}{n!}P\left\{n+1-\sum _{i=1}^k{Y}_i=i_{k+1}:Y_j=i,j=1,2,....k\right\} \\ =\frac{k!\left(n-k\right)!}{n!}, \left\{\sum _{j=1}^{k+1}{i}_j=n+1\right\} \\ =P(y_1,y_2,....y_n) y_i=0,1, \sum _{i=1}^n{y}_i=k \end{gathered}$$

As a result, the joint mass function is symmetric, proving the conclusion. That is to say, the random variables are exchangeable.