1. Review of Probability

2. Definitions (1)

3. Quiz 1.Let’s say I have a random variable X for a coin, with event space {H, T}. If the probability P(X=H) is 0.5, what is P(X=T)? 2. If P(X=H)=0.25, what is P(X=T)?

4. Definitions (2) Joint distribution: A probability distribution of two or more random variables. The event space for this distribution is the cross product of the event space of the individual random variables. E.g. Let X 1 be a random variable for a coin flip. Let X 2 be a random variable for a second coin flip. P(X 1, X 2 ) is a joint distribution over all possible values for both coin flips.

5. Quiz How many events are in the event space for flipping two coins? Name two of these events.

6. Definitions (3) Marginal distribution: This is just any probability distribution, but people use it to refer to a distribution over one variable when they’ve separately introduced a joint distribution over that variable and a second variable. E.g., if I have a joint distribution P(X 1, X 2 ), then P(X 1 ) is a marginal distribution over X 1, and P(X 2 ) is a marginal distribution over X 2.

7. Important Equation

8. Definitions (4)

9. Important Equation (2)

10. Definitions (5)

11. Quiz 1.Suppose I flip a coin 3 times. Each time has P(H)=0.5. Assume the three coin tosses are independent. What is P(H, H, H)? 2.Suppose I flip the coin 4 times, and let the random variable for the i-th time be Xi. What is P(X1=X2=X3=X4)? 3.What is the probability that, in 4 coin flips, I get at least 3 heads?

12. Quiz Suppose I have one normal coin (P(X1=H)=0.5), and one weird coin with the following properties: P(X2=H|X1=H)=0.9 P(X2=T|X1=T)=0.8 If I flip X1 and then X2, what is P(X2=H)?

13. Important Equations so far

14. Quiz: Diabetes P(D)=0.01 (called the prior probability) Test for diabetes is either + or – P(+|D)=0.9 P(-|  D)=0.8 P(-|D) = P(+|  D) = P(+, D) = P(-, D) = P(+,  D) = P(-,  D) = P(D|+) =

15. Bayes Rule Likelihood Prior Marginal Likelihood Posterior

16. Bayes Rule for Diabetes Test P(D|+) = ?