Q.6.18 - A First Course in Probability

Question

Suppose X and Y are both integer-valued random variables. Let p(i|j) = P(X = i|Y = j) and q(j|i) = P(Y = j|X = i) Show that P(X = i, Y = j) = p(i|j)  i p(i|j) q(j|i 

Step-by-Step Solution

Answer

The required expression is $\frac{P(X=i, Y=j)}{\sum _i{\displaystyle \frac{P(X=i, Y=j)}{P(Y=j, X=i)}}}=\frac{p(i/j)}{\sum _i{\displaystyle \frac{p(i,j)}{q(j,i)}}}$

Step 1: Content Introduction

A random variable is a variable that has a set of possible values and probability. It's a variable whose value is determined by the outcome of an unknown event.

Step 2: Content Explanation

We have

$$\begin{gathered} P(X=i, Y=j)=\frac{P(X=i, Y=j)}{1} \\ =\frac{P(X=i, Y=j)}{\sum _{i}P(X=i)} \\ =\frac{P(X=i, Y=j)}{\sum _i{\displaystyle \frac{P(X=i)}{P(Y=j)}}.P(Y=j)} \\ =\frac{P(X=i, Y=j)}{\sum _i{\displaystyle \frac{P(X=i, Y=j)}{P(Y=j, X=i)}}}=\frac{p(i/j)}{\sum _i{\displaystyle \frac{p(i,j)}{q(j,i)}}} \end{gathered}$$