Conditional expectation

1. Overview

In probability theory, the conditional expectation of a random variable is its expected value – the value it would take “on average” over an arbitrarily large number of occurrences – given that a certain set of "conditions" is known to occur

2. Description

2.1 Formula

ith two random variables, if the expectation of a random variable X is expressed conditional on another random variable Y without a particular value of Y being specified, then the expectation of X conditional on Y, denoted:

$$E[X|Y]$$

, is a function of the random variable Y and hence is itself a random variable.

3. Example

Consider the roll of a fair die and let A = 1 if the number is even (i.e. 2, 4, or 6) and A = 0 otherwise. Furthermore, let B = 1 if the number is prime (i.e. 2, 3, or 5) and B = 0 otherwise.

	1	2	3	4	5	6
A	0	1	0	1	0	1
B	0	1	1	0	1	0

• The unconditional expectation of A is $E[A]=\frac{0+1+0+1+0+1}{6}=\frac{1}{2}$.
• the expectation of A conditional on B =1 is $E[A|B=1]=\frac{1+0+0}{3}=\frac{1}{3}$
• The expectation of A conditional on B = 0 is $E[A|B=0]=\frac{0+1+1}{3}=\frac{2}{3}$
• The expectation of B conditional on A = 1 is $E[B|A=1]=\frac{1+0+0}{3}=\frac{1}{3}$

4. Reference

https://en.wikipedia.org/wiki/Conditional_expectation