Statistics 103: Midterm 1 Test Form A (Dated: April 15, 2016) Instructions: Before you do anything, make sure your name, ID and your test form type is on your scantron. You have until noon to finish this test. You may use one sheet of notes and a calculator. 1. Is the following function a probability mass function for some random variable? 0.6 if x = 1 0.2 if x = 0 p(x) = 0.2 if x = 1 0 otherwise (a) Yes Answer: 1a (b) No X be a random draw from the following box Y of tickets: 2. Let 0 0 1 1 2 1 2 0 1 0 5 tickets Find E(X Y ) (a) 0 (c) 8/5 (e) 4/5 Answer: 2a (b) 6/5 (d) 10/5 6. Suppose X is a random variable with probability mass function ( \u0001x 2 13 if x = 1, 2, 3, . . . p(x) = 0 otherwise Find P (X 2). (a) 0.8889 (c) 0.1111 (e) 0.3333 Answer: 6a (b) 0.5556 (d) 0.2222 7. Suppose X and Y are the random variables with joint PMF given by: y x\u001f 3. Let Y be a random random variable with PMF given by: 0.7 if y = 2 0.25 if y = 0 pY (y) = 0.05 if y = 1 0 otherwise Find P (Y = 0 or Y = 1). (a) 0.75 (b) 0.95 (c) 0.0125 (d) 0.25 (e) 0.3 Answer: 3e 4. Suppose X and Y are the random variables with joint PMF given by: y x\u001f 1 0 1 5. Consider a box full of tickets, where 79% of the tickets have a \"1\" written on it, the rest have a \"0\". I randomly choose one ticket and let X be the ticket number I get. What is var(X). (a) 0.5 (b) 0.166 (c) 0.79 (d) 0.1659 (e) 0.2099 Answer: 5d 1 0 1 0.1 0.3 0.1 0.2 0.0 0.1 0.0 0.1 0.1 Compute P (X 0|Y 0). (a) 1/10 (b) 6/7 (c) 3/10 (d) 1/2 (e) 3/7 Answer: 4e 1 0 5 0 0.2 0.15 0.1 1 0.0 0.1 0.15 2 0.2 0.1 0.0 Find E(Y 2 ). (a) 1.35 (c) 6.65 (e) 1.45 Answer: 7c (b) 5.9275 (d) 0.85 8. Referring to the same joint PMF giving is question 7, which is the conditional PMF of X given Y = 0. 0.45 if x = 0 (a) p(x|Y = 0) = 0.25 if x = 1 0.3 if x = 2 0.15 if x = 0 (b) p(x|Y = 0) = 0.1 if x = 1 0.1 if x = 2 15/35 if x = 0 (c) p(x|Y = 0) = 10/35 if x = 1 10/35 if x = 2 15/35 if x = 0 (d) p(x|Y = 0) = 1/35 if x = 1 1/35 if x = 2 2 Answer: 8c Pn Pn 9. Suppose n1 i=1 x2i = 30, n1 i=1 yi2 = 27 and Pn Pn 1 1 2 i=1 xi yi = 22. Find n i=1 (xi + yi ) . n (a) 79 (b) 57 (c) 35 (d) 101 (e) 1673 Answer: 9d Pn Pn 2 10. Suppose i=1 xi = 118, i=1 xi = 750 and Pn n = 25. Find )2 . i=1 (xi x (a) 25.28 (c) 321002 (e) 341.64 Answer: 10b (b) 193.04 (d) 245.91 11. Suppose X and Y are the random variables with joint PMF given by: y x\u001f 0 1 2 0 0 1/5 1/5 0 1 2/15 4/15 2 1/5 0 0 Which of the following four boxes gives a box model X represents a random ticket for X and Y where Y drawn the box. 3 3 2 6 1 0 0 1 1 2 1 2 0 1 0 Box Model 1 (15 tickets total) 3 3 2 4 3 0 0 1 1 2 1 2 0 1 0 Box Model 2 (15 tickets total) 2 2 1 4 1 0 0 1 1 2 1 2 0 1 0 Box Model 3 (10 tickets total) 3 3 2 4 3 1 2 0 1 0 0 0 1 1 2 Box Model 4 (15 tickets total) Note: the number on the top of each ticket tells you how many copies of that ticket are in the box. (a) Box Model 1 (c) Box Model 3 Answer: 11b (b) Box Model 2 (d) Box Model 4 12. Suppose X is a random variable with probability mass function .1 if x = 2 .25 if x = 0 .5 if x = 1 p(x) = .1 if x = 2 .05 if x = 3 0 otherwise Find E(X|X 2). (a) 1.0 (c) 0.15 (e) 2.3333 Answer: 12e (b) 0.35 (d) 2.1