2 distributions - Probability Distributions Waiting (time/#...

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Probability Distributions • Waiting (time/# of draws until event) – Geometric (discrete) – Exponential (continuous) • Counting (how many occurrences) – Binominal – Poisson (rare events approximation) – Normal (Gaussian) -- for continuous values Waiting for an event to occur • Let p = probability of an event (say a mutation) occurring on each trail • Want to compute the time (or number of trails) until the event – Example: How many individuals do we need to score to find a mutation when the mutation rate is (say) 1/1000? – Might expect that this is 1000 individuals, but in reality it’s a bit more complex • Probability first n are failures, next is a success is just (1-p) n p The geometric distribution • The waiting time (number of trails) until a success is often given by the geometric distribution: – Pr(first success on trail n+1) = (1-p) n p • What about a success (i.e., AT LEAST one) in the first k? – Prob(one, or more, successes in first k) – = 1 - Pr(no successes in first k) = 1 - (1-p) k Example: Screening mutations • Suppose mutation rate p = 1/1000. • How many individuals do we need to score to have a 50% chance of seeing at least one mutation? • Solve for k such that 1 - (1-p) k = 0.5 – Rearranging gives -(1-p) k = 0.5 -1 = -0.5 – Hence (1-p) k = 0.5, or k log(1-p) = log(0.5) – Thus k = log(0.5)/log(1-p) – = log(0.5)/log(1-1/1000) = 693. – Prob(success by 1000) = 1-(1-1/1000) 1000 = 0.63
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Class problems • What is prob(at least 1 mutation) for 500 trails? • 1-(1-1/1000)
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This note was uploaded on 12/09/2009 for the course ECOL 320 taught by Professor Weinert during the Spring '07 term at University of Arizona- Tucson.

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2 distributions - Probability Distributions Waiting (time/#...

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