STAT 230 Final, Summer 2010

American University of Beirut STAT 230 Introduction to Probability and Random Variables Summer 2010 Final Exam 1. Let X and Y be a couple of random variables with joint pdf f(x, y) = c , 0 < x < 1 , x2 < y < 1 Find the value of the constant c. 2. Refer to question 1, find E(X). 3. Lori just bought a new set of 4 tires for her car. The life of each tire is normally distributed with a mean of 45000 miles and a standard de- viation of 3200 miles. Find the probability that all 4 tires will last at least 46000 miles. Assume the life of each of these tires is independent of the lives of other tires. 4. A television store owner figures that 45 per- cent of the customers entering his store will purchase on ordinary television set, 15 percent will purchase a plasma television set, and 40 percent will just be browsing. If 5 customers enter his store on a given day, what is the prob- ability that he will sell exactly 2 ordinary sets and 1 plasma set on that day ? 5. A study found that 80% of students are too young to vote. Find the probability that at least 175 students can vote in a random sample of 800 students. 6. Suppose that in a community the distributions of heights of men and women (in centimeters) are N(173, 40) and N(160, 20), respectively. Cal- culate the probability that the average height of 10 randomly selected men is at least 5 cen- timeters larger than the average height of six randomly selected women. 7. Let (X, Y ) be a couple of random variables with joint pdf f(x, y) = e−(x+y) , x > 0 , y > 0 Let U = X + Y and V = X/Y . a. Find the joint pdf of the couple (U, V ). b. Find the marginal pdf of U and V . 8. Otto is trying out for the javelin throw to com- pete in the olympics. The lengths of his javelin throws is normally distributed with a mean of 290 feet and a standard deviation of 10 feet. Find the probability that the longest of three of his throws is 320 feet or more. 9. The time it takes for a student to finish an aptitude test (in hours) has the pdf f(x) = 6(x − 1)(2 − x) , 1 < x < 2 Approximate the probability that the average length of time it takes for a random sample of 15 students to complete the test is less than 1 hour and 25 minutes.