Probability - Caltech

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Random variables ! Events are cumbersome to work with. ! Let D be some set (e.g., the integers) ! A random variable X is a mapping ! For some , we say “probability that variable X assumes state x” 12
Examples ! Bernoulli distribu)on: “(biased) coin flips” D = {H,T} Specify P(X = H) = p. Then P(X = T) = 1-‐p. Note: can iden)fy atomic events ω with {X=H}, {X=T} ! Binomial distribu)on counts the number of heads ! Categorical distribu)on: “(biased) m-‐sided dice” D = {1,…,m} Specify P(X = i) = p i , s.t. ∑ ι p i = 1 ! Mul)nomial distribu)on counts the number of outcomes for each side 13
Page 1 and 2: Introduc)on to Ar)ficial Intellig
Page 3 and 4: Announcements ! Grading: AvianAske
Page 5 and 6: Quan)fying Uncertainty ! So far,
Page 7 and 8: Decision making under uncertaint
Page 9 and 10: Why use probabili)es? Related ev
Page 11: Interpreta)on of probabili)es ! P
Page 15 and 16: Example: Gaussian distribu)on !
Page 17 and 18: Example: Mul)variate Gaussian 17
Page 19 and 20: Marginal distribu)ons What is P(
Page 21 and 22: Condi)onal distribu)ons ! Formal
Page 23 and 24: Posterior inference ! Suppose we
Page 25 and 26: Independent RVs ! What if RVs a

Probability - Caltech

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