need only answers for the assignment
STOR 435 Final Exam SS1 2020 Name: ID#: I pledge that I have neither given nor received unauthorized aid on this exam. Signature: Note: Each of parts (1a) – (1f) counts one point; every other part counts two points. The full score is 40 points. (1) Check “true” or “false”. (a) If X follows Poisson (1), then 3X follows Poisson (3). true ; false . (b) If X and Y have the same distribution, then X + Y and X − Y are uncorrelated. true ; false . (c) Let X and Y denote the total numbers of red balls and white balls respectively when sampling 5 balls without replacement from a box that contains 10 red balls and 5 white balls. Then Cov(X,Y ) = −V ar(Y ). true ; false . (d) If X follows the distribution N(1, σ2), so does 2−X. true ; false . (e) If (X,Y ) is uniformly distributed over the unit disk {(x, y) ∈ IR2 : x2 + y2 ≤ 1}, then P (X + Y > 0) = 2/π. true ; false . (f) If (X,Y ) is uniformly distributed over the square [−1, 1]2, then X and Y are uncorrelated. true ; false . (2) Let X and Y be iid N(0, 1) RV’s. Define U = 2X + Y and V = X − 2Y . 1 yannnsama yannnsama (a) Do U and V have the same distribution? If so, specify the (common) distribution. If not, explain why. (b) Choose one of the following answers for the joint density of (U, V ): (A1) 12π e − 1 2 (u2+v2); (A2) 110π e − 1 10 (u2+v2); (A3) 1√ 10π e− 1 10 (u2+v2); (3) Suppose X has the cdf F (x) = 0, x < −1;="" 1="" 3x+="" 1="" 3="" ,="" −1="" ≤="" x="">< 0;="" 1="" 3x+="" 2="" 3="" ,="" 0="" ≤="" x="">< 1;="" 1,="" x="" ≥="" 1.="" (a)="" draw="" the="" graph="" of="" f="" (x).="" (b)="" choose="" one="" of="" the="" following="" answers="" for="" median="" of="" x:="" -1/2;="" 1/2;="" 0.="" (c)="" choose="" one="" of="" the="" following="" answers="" for="" p="" (|x|=""> 1/2): 1/6; 1/3; 1/2. 2 (d) Choose one of the following answers: (A1) X does not have a density function; (A2) X has a density f(x) with f(0) = 1/3; (A3) Neither (A1) nor (A2) is correct. (e) Choose one of the following answers: (A1) X does not have atoms; (A2) X has an atom 0 with P (X = 0) = 1/3; (A2) X has an atom 0 with P (X = 0) = 1/2. (4) Suppose events A and B are mutually exclusive. Event C is independent of A, and C is also independent of B. Assume P (A) = 1/4, P (B) = 1/3 and P (C) = 1/2. Let N denote the total number of events among A, B, C that occur. (a) Draw a Venn diagram by filling each disjoint part with a correct probability. (b) Choose one of the following answers for EN : 13/12; 2; 5/2. (c) Choose one of the following answers for E(N2): 13/12; 5/3; 11/6. (d) Choose one of the following answers for Cov(IA, N): 0; 11/24; 5/48. (e) Choose one of the following answers for P (N ≤ 2 | C): 1; 5/24; 7/24. 3 (5) Let X and Y be iid each being uniformly distributed over the interval [0, 1]. Define U = X + Y and V = X − Y . Note: The joint density for (U, V ) is f(u,V )(u, v) = 1/2, ∀ (u, v) ∈ D, where the domain D = {(u, v) : 0 ≤ u+ v ≤ 2, 0 ≤ u− v ≤ 2} is the square on the U/V plane whose four vertices have coordinates (0, 0), (1,−1), (2, 0), (1, 1). However, you do not really need this joint density for the following parts (5a) — (5d). (a) Calculate E(UV ). (b) Calculate Cov(U, V ). (c) Calculate corr(2V,X) (d) Calculate the conditional expectation E(UV | X = 1/2). 4 (6) Suppose the number of phone calls per hour arriving at an answering service follows a Poisson process with the rate λ = 60 (or equivalently, the interarrival times are iid exponential random variables with mean 1 minute). Let T (i, j) denote the time interval from the ith arrival to the jth arrival. Circle the correct answer among the following choices: The correlation between T (10, 50) and T (20, 60) is equal to 1/2, 3/4, −1 5