Q.6.13 - A First Course in Probability

Question

In Example 5e, we computed the conditional density of a success probability for a sequence of trials when the first n + m trials resulted in n successes. Would the conditional density change if we specified which n of these trials resulted in successes? 

Step-by-Step Solution

Answer

The conditional density is still beta with parameters $n+1, m+1$

Step 1: Content Introduction

Let define the outcome in ith trail.

X_i = { $1$, if trial is a success and $0$ otherwise

Step 2: Content Explanation

Let X probability that given trial is a success, then X is a uniform $(0,1)$ random variable. The conditional density of X given that N is,
$$\begin{gathered} f_{X|X_1},..{.}_{X_{n+m}}\left(x\right| x_1,....x_{n+m})=\frac{P\{N=n|X=x\} f_x(x)}{P\{N=n\}} \\ =\frac{{\displaystyle \left(\begin{array}{c}n+m\\ n\end{array}\right)}{x}^{\sum _{x_1}}{(1-x)}^{(n+m)-\sum _{x_1}}}{P\{N=n\}} \\ =c\times x^{\sum _{x_i}}(1-x{)}^{(n-m)\underset{x_1}{-\sum }} \end{gathered}$$

where c does not depend on x.