PSTAT W120A, Summer 2016 Assignment 5.B Due Monday Sep. 5 Directions: Please write up your solutions and submit them on GauchoSpace by the end of the day on Monday. We prefer the submission to be single pdf file if possible. Bonus points will be awarded for organization and neatness. Lesson 19 - Independent Normals 1. We have 55 independent normal observations all with mean 100. The first 50 observations have variance 76.4, and the last five have variance 127. (a) Calculate the probability that the first observation X1 is between 98 and 103. = 1 P55 Xi is between 98 and 103. (b) Calculate the probability that the average X 55 i=1 2. Suppose that X and Y are independent standard normal random variables. Find the joint distribution of R = 3X + 2Y and S = 2X 3Y . Hint: Are R and S independent? 3. We have 16 independent normal random variables X1 , . . . , X16 where EXt = t and the variance of each is 2 . We calculate new random variables St = (1 + 3)Xt2 (3 + 3)Xt1 + (3 3)Xt (1 3)Xt+1 for t = 3, . . . , 15. (a) Show that ESt = 0 for any t. (b) Prove S = 12 [X1 + X2 ] and T = T 0 = X1 X2 independent? 1 2 [X1 X2 ] are independent. Are S 0 = X1 + X2 and (c) Calculate 7 X 1 2 E S(2j+1) 7 j=1 Lesson 20 - Discrete Conditional Distributions 4. If I roll a pair of dice and X = the larger of the two rolls and S = the sum of the two rolls, then calculate the conditional probability that X = 4 given that S = 7. 5. Suppose I have a box of 4 red marbles and 4 green marbles and I draw 4 marbles without replacement. Let Y be the number of red marbles among the first two that are drawn. Let X be the total number of red marbles drawn. Find the conditional distribution of X given that the Y = 1. In particular, complete the following table x PX|Y (x | 1) 0 1 2 3 4 1 of 2 PSTAT W120A, Summer 2016 Assignment 5.B Due Monday Sep. 5 Questions 6. In a particular town, the distribution of X =the number of children in each family is x P (x) 0 0.1 1 0.4 2 0.3 3 0.2 Let G be the number of girls in each family. We will suppose that all the children are equally likely to be girls or boys. (a) What is the probability that G = 2 conditional on X = 3? (b) Calculate the marginal probability that G = 2. (c) Find the marginal probability distribution function for G. (d) Calculate E(G). 7. The pair (X, Y ) are distributed according to the joint density f (x, y) = 2x(y x)2 for 0 x 1 and 0 y 2 (a) Calculate P{Y 1} (b) Calculate the marginal density of X. Y (c) Calculate E X . (d) Are X and Y independent? Why or why not? Group Question Please post your solutions to this problem in the online discussion forum. 8. Suppose I have a 1 meter long stick that I break randomly in two places such that the location of the breaks are independently distributed uniformly along the length of the stick. (a) Calculate the probability that one of the pieces will be longer than 75 cm. (b) Calculate the probability that the three resulting pieces can be arranged to form a triangle. 2 of 2