Question

It is March, and you take bus to go to school every day. If the weather is clear, it takes exactly 30 minutes to go to school with the bus. However, if the weather is bad, travel time of the bus becomes a random variable T = 30 + D, where D is the delay due to bad weather. The weather condition is a random variable too, denoted by W. Table 1 shows sample measurements for the weather in March obtained from past years weather reports, where lower value for W means clear weather and higher values corresponds to worse weather conditions. Table 2 shows the samples from the delay in travel time of the bus (in minutes) from the days with bad weather. Two tables are collected independently and all measurements in the individual tables are also independent. 2 Table 1: Samples for Weather Condition W Sample Number w 1 0.3572 2 0.7586 3 0.0915 4 0.6365 5 0.2454 6 0.3995 7 0.6658 8 0.8117 9 0.1647 10 0.8782 Table 2: Samples for delay in bus travel time D under bad weather conditions. Sample Number d 1 25.6645 2 37.6529 3 18.1239 4 17.8418 5 22.4293 6 169.6558 7 133.8231 8 57.0343 9 31.6624 10 21.1983 a Assume that weather condition W can be modeled with a normal distribution with mean µ and standard deviation σ. By using Table 1, find the maximum likelihood estimates, µˆ and σˆ (4 points) b Assume that bad weather in the problem definition corresponds to the score W ≥ 0.65. Using the distribution from the previous part, calculate the probability of experiencing bad weather on a given day. You can use the tables of standard normal distribution (see next page) to find this value. (5 points) c Assume that the delay in bus’ travel time D when the weather is bad, can be modeled with an exponential distribution with parameter λ. Using data from Table 2, compute the maximum likelihood estimate λˆ. (4 points) d Using the results from all previous steps, find the PDF, CDF, expectation and variance of the travel time of the bus T. (12 points) e Let N be a random variable that represents the number of bad weather days in March. Use the Central Limit Theorem to compute the probability that N > 10. (10 points)

Answer 1