Chapter 2.7 - Math3200 Intermediate Probability and Statistics Prof Nan Lin Department of Mathematics Washington University Outline Bernoulli Binomial

Chapter 2.7 - Math3200 Intermediate Probability and...

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Math3200 Intermediate Probability and Statistics Prof. Nan Lin Department of Mathematics Washington University
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Outline Bernoulli Binomial Hypergeometric Poisson Geometric Multinomial
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Bernoulli(p) Bernoulli trials are random experiments where outcome is classified into one of 2 mutually exclusive and exhaustive categories. e.g., heads vs. tails, winning vs. losing, graduating vs. dropping out, etc. ? and 1 − ? are the probabilities of success and failure, respectively. Suppose X (success) = 1 and X (failure) = 0. The p.m.f. is, ? ? = ? 𝑥 1 − ? 1−𝑥 , ? = 0,1
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Bernoulli(p) Mean: ? Variance: ?(1 − ?) Lottery example : Suppose the probability of winning the lottery is .1. What is the probability of buying two losing tickets, a winning ticket, and then two losing tickets? Let A = {0, 0, 1, 0, 0} and assuming independence implies, 𝑃 ? = .9 .9 .1 .9 .9 = .1 .9 4 = 0.066 Note that 𝑃(?) is the same for any combination of 1 win and 4 losses.
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Binomial(n,p) When a sequence independent Bernoulli trials occur, we are often more interested in the total number of successes and not the order of occurrences. Let ? equal the number of successes in ? independent Bernoulli trials If ? successes occur then ? − ? failures occur and the number of unique arrangements is ? ? . The p.m.f. of ? is ? ? = ? ? ? 𝑥 1 − ? ?−𝑥 , ? = 0,1,2, … , ?
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Binomial(n,p) Assumptions ? Bernoulli trials are performed (i.e., ? is fixed). All trials are independent. The probability of success (and failure) is constant. ? is the number of successes in ? trials. When these assumptions are met, we say ?~?𝑖???𝑖𝑎?(?, ?)
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Mean of Binomial(n,p) Substituting ? = ? + 1 to yield, Binomial(n-1,p)
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Variance of Binomial(n,p) Substituting x=k+2 to yield Then
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From Bernoulli to Binomial If ? 1 , ? 2 , … , ? ? are independently and identically distributed (i.i.d.) as Bernoulli(p), then ? = ? 1 + ? 2 + ⋯ + ? ? ~?𝑖???𝑖𝑎? ?, ? So, 𝐸 ? = 𝐸 ? 1 + ⋯ + 𝐸 ? ? = ?? 𝑉𝑎? ? = 𝑉𝑎? ? 1 + ⋯ + 𝑉𝑎? ?
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