slides11 - Lecture Stat 302 Introduction to Probability -...

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Lecture Stat 302 Introduction to Probability - Slides 11 AD March 2010 AD () March 2010 1 / 19
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Discrete Random Variables A discrete r.v. X takes at most a countable number of possible values f x 1 , x 2 , ... g with p.m.f. p ( x i ) = P ( X = x i ) where p ( x i ) 0 and i = 1 p ( x i ) = 1 . Expected value/mean μ = E ( X ) = i = 1 x i p ( x i ) . Variance Var ( X ) = E ( X ± μ ) 2 ± = E ² X 2 ³ ± μ 2 . AD () March 2010 2 / 19
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Poisson Random Variable A discrete r.v. X taking values 0 , 1 , 2 , ... is said to be a Poisson r.v. with parameter λ , λ > 0, if p ( i ) = P ( X = i ) = e λ λ i i ! . This expresses the probability of a number of events occurring in a known average rate λ . If we consider a binomial r.v. X of parameters ( n , p ) such that n is large and p is small enough so that np is moderate then the binomial distribution can be well-approximated by the Poisson distribution of parameter λ = np . AD () March 2010 3 / 19
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Example: Waiting at the bus stop Example : Assume that 6 buses per hour stop at your bus stop. If the buses were arriving exactly every 10 minutes, then your average waiting time would be 5 minutes; i.e. you wait between 0 and 10 minutes. What is the probability that you are going to wait at least 5 minutes without seeing any bus if the buses follow a Poisson distribution? What is the proba to wait at least 10 minutes? What is the proba of seeing two buses in 10 minutes? Answer : If we let X 1 be the number of buses arriving in 5 minutes, it is a Poisson r.v. with parameter 0 . 5 (average rate 6 per hour). So we have P ( X 1 = 0 ) = e & 0 . 5 = 0 . 60 . If we let X 2 be the number of buses arriving in 10 minutes, it is a Poisson r.v. with parameter 1. So we have P ( X 2 = 0 ) = e 1 = 0 . 368 , P ( X 2 = 2 ) = e 1 1 2 2 ! = 0 . 184 . AD () March 2010 4 / 19
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We have E ( X ) = λ . We have E ( X ) = i = 0 i P ( X = i ) = i = 0 i e λ λ i i ! = e λ i = 0 λ λ i 1 ( i 1 ) ! = e λ λ j = 0 λ j j ! (change j i 1) = e λ λ e λ = λ . Note that this is in agreement with our approximation of Binomial by Poisson. A Binomial has mean np and we approximate it by a Poisson of parameter λ = np which is also the mean of the Poisson distribution. AD ()
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slides11 - Lecture Stat 302 Introduction to Probability -...

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