MA 320 Introductory Probability, Section 1 Spring 2017 Final Exam Practice May. 1, 2017 Exam Scores Question Score Total 1 10 Name: Last 4 digits of student ID #: No books or notes may be used. Turn off all your electronic devices and do not wear ear-plugs during the exam. You may use a calculator, but not one that has symbolic manipulation capabilities or a QWERTY keyboard. Free Response Questions: Show all your work on the page of the problem. Show all your work. Clearly indicate your answer and the reasoning used to arrive at that answer. 2 10 3 10 4 10 5 10 6 10 7 10 8 10 9 10 10 10 Total 100
1. (10 points) Let X and Y be independent discrete random variables with the following probability distributions: x P (X = x) 1 5/13 2 8/13 y P (Y = y) 1 1/3 2 2/3 A. (1 point) Find the expected value of X. B. (2 points) Find the expected value of X 2. C. (1 point) Compute P (X + Y = 2). D. (1 point) Compute P (X + Y = 3). E. (2 points) Compute P (X + Y = 3 and Y = 1). F. (3 points) Compute P (X + Y = 3 given Y = 1). 2
2. (10 points) Let X 1,..., X n be an independent trials process with common mean and variance E(X i ) = 2, V (X i ) = 7. A. (2 points) Use E(X i ) and V (X i ) to find E(X 2 i ). B. (2 points) Let S n = X 1 + X 2 +... + X n be the sum of all the variables. Find the expected value of S n as a formula of n. C. (2 points) Find the variance of S n as a formula of n. D. (2 points) Let A n = S n n of A n. be the average of the variables. Find the expected value E. (2 points) Find the variance of A n as a formula of n. 3
3. (10 points) Let X 1,..., X n be an independent trials process with common mean and variance E(X i ) = 2, V (X i ) = 7. A. (2 points) What (whole number) does the Chebyshev inequality give as a bound on P ( X i E(X i ) 1/4)? B. (3 points) Can P ( X i E(X i ) 1/4) equal your answer to part (A)? If you write yes, give an example. If you write no, explain why based on how large the probability of any event can be. C. (2 points) What does the Chebyshev inequality give as a bound on Your answer should be a formula of n. P ( A n E(A n ) 1/4)? D. (3 points) Using the bound from part (D), what is the minimum number of trials n needed so that P ( A n E(A n ) 1/4) 10/11? Round your answer up to the next integer. 4
4. (10 points) Let X be a continuous random variable in [42, 84] with the following density function: 1/42 42 x 84, f X (x) =. 0 otherwise A. (2 points) Compute the expected value of X. B. (3 points) Compute the expected value of X 2. C. (2 points) Let φ(x) = 1 42 x 63, 7 otherwise. Compute the expected value of φ(x). D. (3 points) Let ψ(x) = 1 42 x 56,. If the expected value of ψ(x) equals 7, c otherwise what number should c equal? 5
5. (10 points) Let X and Y be independent continuous random variables with density functions f X (x) and f Y (y) given by: f X (x) = 3x 2 0 x 1, f Y (y) = 4y 3 0 y 1, A. (5 points) Let φ(x, y) = xy 2 Find the expected value of φ(x, Y ). Give an exact answer using fractions. B. (5 points) Let ψ(x, y) = 1 0 y x, 2 x y 1, Find the expected value of ψ(x, Y ). Give an exact answer using fractions. 1 Y 0 0 X 1 6
6. (10 points) Let X and Y be independent continuous random variables with density functions f(x) and g(y) given by: f(x) = 1 2 e x/2, if x 0 g(y) = 1 2 e y/2, if y 0 In this problem, you will find the density function (f g)(z) of Z = X + Y for z 0 by computing (f g)(z) = f(z y) g(y) dy A. (3 points) The function f(z y) is non-zero when y is in what interval? Your endpoints should have z. Reminder: a y b means y is in the interval [a, b]. B. (2 points) Let z 0. On what interval is f(z y) g(y) non-zero? C. (5 points) Set up the integral for computing (f g)(z) and then evaluate. Your answer should be a formula with z. 7
7. (10 points) Let X and Y be independent continuous random variables with density functions f X (x) and f Y (y) given by: f X (x) = 1 2 0 x 2, f Y (y) = 1 2 0 y 2, A. (5 points) Find the density function, f(z), of Z = X + Y for 0 z 2. Your answer should be a function of z. B. (5 points) Find the density function, f(z), of Z = X + Y for 2 z 4. Your answer should be a function of z. 8
8. (10 points) In this problem, let Z have the standard normal distribution and S 100 be a sum of 100 random variables (from an independent trials process) with E(S 100 ) = 26, V (S 100 ) = 100 (note that these are for the sum, not the individual variables) The following tables gives some probabilities for P (0 Z d) to three decimal places: d P (0 Z d) 0.25 0.099 0.5 0.192 0.75 0.273 1 0.341 d P (0 Z d) 1.25 0.394 1.5 0.433 1.75 0.460 2 0.477 d P (0 Z d) 2.25 0.488 2.5 0.494 2.75 0.497 3 0.499 A. (2 points) Find P ( 1.5 Z 0). Please give three decimal places. B. (3 points) Find P ( 1.5 Z 1). Please give three decimal places. C. (5 points) Estimate P (18.5 S 100 53.5) using the CLT. Please give three decimal places. 9
9. (10 points) Let X 1,..., X n be an independent trials process. Suppose that 100 trials are conducted, giving a sample mean of µ = 46 and a sample variance of σ 2 = 81. Compute the 95% confidence interval for this set of trials. You may use fractions or exact decimals. A. (5 points) The 68% confidence interval is (, ). B. (5 points) The 95% confidence interval is (, ). 10
10. (10 points) Let X 1,..., X n be an independent trials process. Suppose that the common variance σ 2 satisfies σ 2 16/9. A. (5 points) What is the minimum number of trials n needed so that the length of the 68% confidence interval is less than 1/2? Round your answer up to the next integer. B. (5 points) What is the minimum number of trials n needed so that the length of the 95% confidence interval is less than 1/2? Round your answer up to the next integer. 11