Fundamentals of Probability and Statistics for Engineers

188 Fundamentals of Probability and Statistics for Engineers6.26 Each air traffic controller at an airport is given the responsibility of monitoring atmost 20 takeoffs and landings per hour. During a given period, the average rate oftakeoffs and landings is 1 every 2 minutes. Assuming Poisson arrivals and departures,determine the probability that 2 controllers will be needed in this timeperiod.6.27 The number of vehicles crossing a certain point on a highway during a unit timeperiod has a Poisson distribution with parameter . A traffic counter is used torecord this number but, owing to limited capacity, it registers the maximumnumber of 30 whenever the count equals or exceeds 30. Determine the pmf of Yif Y is the number of vehicles recorded by the counter.6.28 As an application of the Poisson approximation to the binomial distribution,estimate the probability that in a class of 200 students exactly 20 will have birthdayson any given day.6.29 A book of 500 pages contains on average 1 misprint per page. Estimate theprobability that:(a) A given page contains at least 1 misprint.(b) At least 3 pages will contain at least 1 misprint.6.30 Earthquakes are registered at an average frequency of 250 per year in a givenregion. Suppose that the probability is 0.09 that any earthquake will have amagnitude greater than 5 on the Richter scale. Assuming independent occurrencesof earthquakes, determine the pmf of X, the number of earthquakes greater than 5on the Richter scale per year.6.31 Let X be the number of accidents in which a driver is involved in t years. Inproposing a distribution for X, the ‘accident likelihood’ varies from driver todriver and is considered as a random variable. Suppose that the conditional pmfp X (xj) is given by the Poisson distribution,p X …kj† ˆ…t†k e t; k ˆ 0; 1; 2; ...;k!and suppose that the probability density function (pdf) of(a, b > 0)8 a a a 1>:0;elsewhere,is of the formwhere(a) is the gamma function, defined by…a† ˆZ 1x a01 e x dx:Show that the pmf of X has a negative binomial distribution in the formp X …k† ˆ…a ‡ k†k! …a†a abta ‡ bt k; k ˆ 0; 1; 2; ...:a ‡ btTLFeBOOK