Probability Review

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Probability Review

• Definitions/Identities
• Random Variables
• Expected Value
• Joint Distributions
• Conditional Probabilities

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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.

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Probability Identities
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Joint Distributions

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

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Two Events
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Random Variables
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Multiple Random Variables
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Joint probability matrix
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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(D15), P(D26)
• P(D11), P(D1 D22)

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Dice Example
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Conditional Probability
A?B
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Example
p(y3) 0.7
p(y2) 0.1
p(y1) 0.2
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Markov Processes

• State transition probabilities
• Matrix or Diagram
• Matrix Multiplication predicts multiple
  transition probabilities
• Mk Converges to steady state