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Department
Statistics
Course
STAT 230
Professor
Changbao Wu
Semester
Winter

Description
1. Let X be a random variable that takes possible values at non-negative integers and has cumulative distribution function ▯x F(x) = P(X ▯ x) = 1 ▯ 2 for x = 0;1;2;▯▯▯ (a) Find the probability function of X. Soln: f(x) = P(X = x) = F(x) ▯ F(x ▯ 1) = 2 ▯(x▯1)▯ 2▯x = 2▯x , x = 1;2;▯▯▯. f(0) = 0, because F(0) = 0. (b) Find P(X = 5) and P(X ▯ 5) ▯5 Soln: P(X = 5) = f(5) = 2 ; P(X ▯ 5) = 1 ▯ P(X ▯ 4) = 1 ▯ F(4) = 1 ▯ (1 ▯ 2 ▯4) = 2▯4. P 1 ▯x ▯4 Or: P(X ▯ 5) = x=52 = 2 . (c) Find P(X = 5 j X ▯ 5) Soln: ▯5 P(X = 5; X ▯ 5) P(X = 5) 2 P(X = 5 j X ▯ 5) = P(X ▯ 5) = P(X ▯ 5) = 2▯4 = 0:5 2. During rush hour the number of cars passing through a particular intersection has a Poisson distri- bution with an average of 540 per hour. (Notes: Numerical simpli▯cations are not required; The phrase \30 second interval" is denoted by \30SI".) [2] (a) Find the probability p that there are 11 cars in a 30 second interval (30SI). Soln: Let X = #cars in a 30 second interval, then X ▯ Poisson(▯), with ▯ = 540=(60▯2) = 4:5. 11 4:5 ▯4:5 p = P(X = 11) = 11! e [2] (b) Find the probability that, when 20 disjoint 30SIs are studied, exactly 2 of them had 11 cars. 4:51 ▯4:5 20 2 18 Soln: Let p = 11!e , the probability is given by ( 2p (1 ▯ p) . [2] (c) Give an expression for the probability that we have to observe 140 30SIs to ▯nd the 12th 30SI which has 11 cars. 11 Soln: Let p = 4:5 e▯4:5, the probability is given by ( 11)p (1 ▯ p) 128. 11! [1] (d) Find the Poisson approximation to the probability in (c) Soln: Note that the probability in (c) is from a negative binomial distribution, not a binomial. Poisson approximation can only be used for binomial probabilities. Let ▯ = 139p, then 139 12 128 139 11 128 : ▯11 ▯▯ ( 11)p (1 ▯ p) = ( 11)p (1 ▯ p) ▯ p = e ▯ p
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