1. Probability Review Definitions/Identities Random Variables
Expected Value
Joint Distributions
Conditional Probabilities

2. Probability Defined
an event (experiment) has a set of possible outcomes, each with a probability, that measures their relative (anticipated) frequencies of occurrence normalized to 1.

3. Probability Identities
Events and outcomes:
Probability of each outcome:
Probability distribution:

4. Joint Distributions Two (or more) events Each event has an outcome
Joint distribution stipulates the probability of every combination of outcomes

5. Two Events

6. Random Variables

7. Multiple Random Variables

8. Joint probability matrix

9. Conditional Probability
Random variables are often NOT independent
P(rain in Pittsburgh), P(rain in Monroeville), P(rain in New York), P(rain in Hong Kong)
P(Heads up), P(Tails down)
P(D1=5), P(D2=6)
P(D1=1), P(D1 + D2=2)

10. Dice Example

11. Conditional Probability
P(A|B) =
P(AB)
P(B)
AB

12. Example
p(y1) = 0.2
p(y2) = 0.1
p(y3) = 0.7

13. Markov Processes State transition probabilities Matrix or Diagram
Matrix Multiplication predicts multiple transition probabilities
Mk Converges to steady state