Binomial distribution

# Binomial distribution - Binomial distribution Number of...

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Binomial distribution 1 / 14 Binomial distribution Number of heads when a coin with heads bias p [0 , 1] is tossed n times: binomial distribution S Bin( n, p ) . Probability mass function : for any k ∈ { 0 , 1 , 2 , . . . , n } , P ( S = k ) = n k p k (1 - p ) n - k . k 0 5 10 15 20 25 30 35 40 45 50 Pr[S=k] 0 0.02 0.04 0.06 0.08 0.1 2 / 14 Special case: Bernoulli distribution The outcome of a coin toss with heads bias p [0 , 1] : Bernoulli distribution X Bern( p ) = Bin(1 , p ) P ( X = 1) = p , P ( X = 0) = 1 - p . Mean : E ( X ) = P ( X = 0) · 0 + P ( X = 1) · 1 = p . Variance : var( X ) = E ( X - E ( X ) ) 2 = p (1 - p ) . (Standard deviation is var( X ) ; more convenient to use than E | X - E ( X ) | .) 3 / 14 Binomial = sums of i.i.d. Bernoullis Let X 1 , X 2 , . . . , X n be i.i.d. Bern( p ) random variables, and let S Bin( n, p ) . Then S has the same distribution as X 1 + X 2 + · · · + X n . Mean : By linearity of expectation , E ( S ) = E n i =1 X i = n i =1 E ( X i ) = np . Variance : Since X 1 , X 2 , . . . , X n are independent , var( S ) = var n i =1 X i = n i =1 var( X i ) = np (1 - p ) . 4 / 14

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Test error rate Let ˆ f : X → Y be a classifier, and suppose you have i.i.d. test data T (that are
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