Conditional Probability & Conditional Expectation

1. Conditional Probability & Conditional Expectation Conditional distributions Computing expectations by conditioning Computing probabilities by conditioning Chapter 3

2. Discrete conditional distributions Given a joint probability mass function the conditional pmf of X given that Y = y is The conditional expectation of X given Y = y is Chapter 3

3. Continuous conditional distributions Given a joint probability density function the conditional pdf of X given that Y = y is This may seem nonsensical since P{Y = y} = 0 if Y is continuous. Interpret as the conditional probability that X is between x and x + dx given that Y is between y and y + dy. The conditional expectation of X given Y = y is Chapter 3

4. Computing Expectations by Conditioning Suppose we want to know E[X] but the distribution of X is difficult to find. However, knowing Y gives us some useful information about X – in particular, we know E[X|Y=y]. • E[X|Y=y] is a number but E[X|Y] is a random variable since Y is a random variable. • We can find E[X] from If Y is discrete then If Y is continuous then Chapter 3

5. Computing Probabilities by Conditioning Suppose we want to know the probability of some event, E (this event could describe a set of values for a random variable). Knowing Y gives us some useful information about whether or not E occurred. Define an indicator random variable Then P(E) = E[X], P(E|Y = y) = E[X|Y = y] So we can find P(E) from Chapter 3

6. Strategies for Solving Problems • What piece of information would help you find the probability or expected value you seek? • When dealing with a sequence of choices, trials, etc., condition on the outcome of the first one Can also find variance by conditioning: Chapter 3