n be the outcome of the i th ip 1 the ith toss gives

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Unformatted text preview: Chen ([email protected]) ENGG2430C lecture 6 14 / 17 Binomial Random Variable Experiment: ﬂip a coin Independently for n times. Deﬁne Bernoulli/indicator r.v.s Xi , (i = 1, 2, . . . , n) be the outcome of the i -th ﬂip: 1, the ith toss gives HEAD; Xi = 0, the ith toss gives TAIL. The PMF is pXi (1) = p and pXi (0) = 1 − p . Let Y = ∑n=1 Xi . Then Y is a Binomial random variable. Its PMF is i given by n pY (k ) = P (Y = k ) = P ∑ Xi = k i =1 = (n ) p k (1 − p )n−k . k Binomial r.v.s model aggregate outcomes of a series of independent experiments. M. Chen ([email protected]) ENGG2430C lecture 6 15 / 17 Binomial Random Variable * Expectation and variance of a Binomial r.v. Y with parameter n and p: Y = ∑n=1 Xi where Xi are independent Bernoulli random variables i The expectation and variance of a Binomial r.v. Y with parameter n and p : n E [Y ] = ∑ E [Xi ] = np , i =1 n Var(Y ) = ∑ Var(Xi ) = np (1 − p ). i =1 M. Chen ([email protected]) ENGG2430C lecture 6 16 / 17 Thank You Reading: Ch. 2.3-2.7 of the textbook. Next lecture: Total expectation theorem, example RV, and application examples. M. Chen ([email protected]) ENGG2430C lecture 6 17 / 17...
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This document was uploaded on 03/31/2014.

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