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Probability Review

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

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Probability Review

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

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

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

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

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