Homework 4: Probability And Statistics

University:
California State University, Northridge
Course:
MATH 340 | Introduction to Probability and Statistics
Academic year:

2023
Views:

471

Pages:

3
Author:

Nyi Moe

11/19/23, 11:37 PM HW04-Nyi Nyi Moe Htet Student: Nyi Nyi Moe Htet Date: 11/19/23 Instructor: Xiaolong Han Course: CSUN_MATH340_HAN (1) Assignment: HW04 1. Three dice are rolled. By assuming that each of the 63 = 216 possible outcomes is equallylikely, find the probabilities attached to the possible values that X can takeon, where X is the sum of the 3 dice. P{X = 1} = 0 P{X = 10} = P{X = 2} = 0 P{X = 11} = P{X = 3} = P{X = 4} = P{X = 5} = P{X = 6} = P{X = 7} = P{X = 8} = P{X = 9} = 1 216 1 72 1 36 5 108 5 72 7 72 25 216 1 8 1 8 25 P{X = 12} = 216 7 P{X = 13} = 72 5 P{X = 14} = 72 5 P{X = 15} = 108 1 P{X = 16} = 36 1 P{X = 17} = 72 1 P{X = 18} = 216 (Simplify youranswers.) 2. Suppose that a 10-sided die is rolled twice. The sides of the die are labeled from 1 to 10. What are the possible values that the following random variables can takeon? (a) the maximum value to appear in the two rolls (b) the minimum value to appear in the two rolls (c) the sum of the two rolls (d) the value of the first roll minus the value of the second roll (a) The possible values of the maximum value to appear in the two rolls are all integers from 1 to 10 . 1 to 10 . (Type integers or decimals. Do not round. Use ascendingorder.) ( b) The possible values of the minimum value to appear in the two rolls are all integers from (Type integers or decimals. Do not round. Use ascendingorder.) ( c) The possible values of sum of the two rolls are all integers from (Type integers or decimals. Do not round. Use ascendingorder.) 2 to 20 . (d) The possible values of the value of the first roll minus the value of the second roll are all integers from −9 to 9 . (Type integers or decimals. Do not round. Use ascendingorder.) https://tdx.acs.pearsonprd.tech/api/v1/print/highered 1/3 11/19/23, 11:37 PM HW04-Nyi Nyi Moe Htet 3. Two coins are to be flipped. The first coin will land on heads with probability 0.2, the second with probability 0.4. Assume that the results of the flips areindependent, and let X equal the total number of heads that result. ( a) Find P{X = 1}. (b) Determine E[X]. (a) P{X = 1} = 0.44 (Type an integer or a decimal. Do notround.) ( b) E[X] = 0.6 (Type an integer or a decimal. Do notround.) 4. If E[X] = 1 and Var(X) = 4, (a) find E (4 + X)2 and (b) find Var(3 + 5X). 5. (a) E (4 + X)2 = 29 (Type an integer or a decimal. Do notround.) (b) Var(3 + 5X) = 100 (Type an integer or a decimal. Do notround.) How many people are needed so that the probability that at least one of them has the same birthday as you is greater than 1 18 ? Assume it is not a leap year and your birthday is not February 29th. 21 people (Type a wholenumber.) 6. This question has been removed from this assignment by your instructor; you have received full credit. 7. There are N distinct types ofcoupons, and each time one is obtained itwill, independently of pastchoices, be of type i with probability Pi , i = 1, ..., N. Let T denote the number one need select to obtain at least one of each type. Compute P{T = n}. Choose the correct answer below. A. P{T = n} = P{T > n − 1} − P{T > n} B. P{T = n} = P{T > n − 1} + P{T < n + 1} C. P{T = n} = 1 − (P{T > n − 1} − P{T > n}) D. P{T = n} = 1 − (P{T > n − 1} + P{T < n + 1}) 8. This question has been removed from this assignment by your instructor; you have received full credit. 9. 2 Let X be a random variable having expected value μ and variance σ . Find the expected value and variance of Y = X−μ σ . Find the expected value. E[Y] = 0 (Simplify youranswer.) Find the variance. ar(Y) = V 1 (Simplify youranswer.) https://tdx.acs.pearsonprd.tech/api/v1/print/highered 2/3 11/19/23, 11:37 PM 10. HW04-Nyi Nyi Moe Htet n A family has n children with probability αp , n ≥ 1, where α ≤ 1−p p . ( a) What proportion of families has nochildren? (b) If each child is equally likely to be a boy or a girl(independently of eachother), what proportion of families consists of k boys(and any number ofgirls)? (a) The proportion of families having no children is 1− α p . 1−p (Simplify youranswer.) ∞ (b) Conditioning on the number ofboys, for k > 0, P{k boys} = ∑ n=k boys} = 1 − α p 1−p ∞ + ∑ n=1 α p n 1 n 1 k 2 n α n p , , and for k = 0, P{0 n . 2 11. This question has been removed from this assignment by your instructor; you have received full credit. 12. Complete parts(a) and(b) to obtain a set of recursive equations for determining Pn , the probability that there is a string of k consecutive heads in a sequence of n flips of a fair coin that comes up heads with probability p. ( a) Argue that for k < n, there will be a string of k consecutive heads if either(1) there is a string of k consecutive heads within the first n − 1 flips, or(2) there is no string of k consecutive heads within the first n − k − 1 flips, flip n − k is atail, and flips n − k + 1, ..., n are all heads. Choose the correct answer below. A. There will be a string of k consecutive heads within the first n trials either if there is one within the first n − k trials, or if the first such string occurs at trial n − k + 1; the former case is equivalent to the conditions of(2). B. There will be a string of k consecutive heads within the first n trials either if there is one within the first n − 1 trials, or if the first such string occurs at trialn; the former case is equivalent to the conditions of(2). C. There will be a string of k consecutive heads within the first n trials either if there is one within the first n − k trials, or if the first such string occurs at trial n − k + 1; the latter case is equivalent to the conditions of(2). D. There will be a string of k consecutive heads within the first n trials either if there is one within the first n − 1 trials, or if the first such string occurs at trialn; the latter case is equivalent to the conditions of(2). k (b) Using thepreceding, relate Pn to Pn − 1 . Starting with Pk = p , the recursion can be used to obtain Pk + 1 , then Pk + 2 , and soon, up to Pn . What is true about cases(1) and(2) in part(a)? A. They encompass all possible outcomes. B. They are equivalent whenever k is less than n 2 . C. They are equivalent in all cases. D. They are mutually exclusive. Use this information to relate Pn to Pn − 1 . Pn = Pn − 1 + 1 − Pn − k − 1 (1 − p)p k 13. This question has been removed from this assignment by your instructor; you have received full credit. https://tdx.acs.pearsonprd.tech/api/v1/print/highered 3/3