Probability Estimation in Exponential Random Variables

STAT/MA 41600 Practice Problems: November 14, 2014 1. Each of the 30 students in a class each orders a package. They assume that the waiting time (measured in days) for the packages are independent exponential random variables, with average waiting time of 1 / 2 for each package. What is the approximate probability that the total waiting time exceeds 14 days? 1

2. When the students in question #1 eventually receive their packages, sometimes they are happy with the items they ordered, and sometimes they are not. Suppose that a stu- dent is happy with her/his own package with probability 0 . 60, independent of the happi- ness/unhappiness of the other students. What is an estimate for the probability that 20 or more of the students are happy with their packages? 2

3. In planning for an event, the planner estimates that nobody will be on time, but nobody will be more than 10 minutes late. So he estimates that the time (in minutes) a given person will be late has density f X ( x ) = (10 - x ) 3 2500 , for 0 ≤ x ≤ 10 , and f X ( x ) = 0 otherwise. a. Find the expected value and variance of X . Hint: It might be helpful to use the u -substitution u = 10 - x . b. Estimate the probability that, among a group of 200 attendees who behave indepen- dently and follow the behavior described above, the total sum of their delay in arriving is more than 420 minutes, i.e., 7 hours. 3

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4. Suppose that 100 marathon runners each complete a marathon in 3.5 hours, on average, with standard deviation 0.5 hours. Estimate the probability that the sum of their completion times is between 348 and 352 hours. 4

5. Barbara is an inspector for a water bottling company. She notices that the amount of water in each bottle has an average of 0 . 99 liters, and a standard deviation of 0 . 03 liters. She measures the quantities X 1 , . . . , X 12 in twelve independent bottles, and computes the average, Y , in these 12 bottles, i.e., Y = X 1 + ··· + X 12 12 . Estimate the probability that Y ≥ 1, i.e., that the average amount of water in the twelve bottles exceeds 1 liter. 5

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