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

2023
Views:

310

Pages:

4
Author:

Nyi Moe

11/19/23, 11:37 PM HW06-Nyi Nyi Moe Htet Student: Nyi Nyi Moe Htet Date: 11/19/23 Instructor: Xiaolong Han Course: CSUN_MATH340_HAN (1) Assignment: HW06 1. Two fair dice are rolled. Find the joint probability mass function of X and Y when ( a) X is the largest value obtained on any die and Y is the sum of thevalues; (b) X is the value on the first die and Y is the larger of the twovalues; (c) X is the smallest and Y is the largest value obtained on the die. (a) Fill in all missing probabilities to complete the table. (Simplify youranswers.) X=x 1 2 1 36 3 4 5 6 Y=y 7 8 9 0 0 0 0 0 0 0 1 1 18 36 0 0 0 0 0 0 0 0 2 0 3 0 0 4 0 0 0 5 0 0 0 0 6 0 0 0 0 1 1 1 18 18 36 1 1 1 1 18 18 18 36 1 1 1 1 18 18 18 18 1 1 1 18 18 18 0 0 (b) Fill in all missing probabilities to complete the table. (Simplify youranswers.) Y=y X=x 1 1 2 3 4 5 6 1 1 1 1 1 1 36 36 36 36 36 36 1 1 1 1 1 18 36 36 36 36 2 0 3 0 0 4 0 0 0 5 0 0 0 0 6 0 0 0 0 1 1 1 1 12 36 36 36 1 1 1 9 36 36 5 1 36 36 0 1 6 (c) Fill in all missing probabilities to complete the table. (Simplify youranswers.) https://tdx.acs.pearsonprd.tech/api/v1/print/highered 1/4 11/19/23, 11:37 PM HW06-Nyi Nyi Moe Htet Y=y X=x 1 1 2 3 4 5 6 1 1 1 1 1 1 36 18 18 18 18 18 1 1 1 1 1 36 18 18 18 18 2 0 3 0 0 4 0 0 0 5 0 0 0 0 6 0 0 0 0 1 1 1 1 36 18 18 18 1 1 1 36 18 18 1 1 36 18 0 1 36 2. Suppose that 3 balls are chosen without replacement from an urn consisting of 4 white and 8 red balls. Suppose that the white balls arenumbered, and let Yi equal 1 if the ith white ball is selected and 0 otherwise. Give the joint probability mass function of (a) Y1 , Y2 ; (b) Y1 , Y2 , Y3 . (a) Find the joint probability mass function of Y1 , Y2 . p(0,0) = 6 11 9 p(0,1) = p(1,0) = p(1,1) = 44 1 22 (Type integers or simplifiedfractions.) (b) Find the joint probability mass function of Y1 , Y2 , Y3 . p(0,0,0) = 21 55 p(0,0,1) = p(0,1,0) = p(1,0,0) = p(0,1,1) = p(1,0,1) = p(1,1,0) = p(1,1,1) = 9 55 9 220 1 220 (Type integers or simplifiedfractions.) https://tdx.acs.pearsonprd.tech/api/v1/print/highered 2/4 11/19/23, 11:37 PM HW06-Nyi Nyi Moe Htet 3. The number of people who enter a drugstore in a given hour is a Poisson random variable with parameter λ = 10. Compute the conditional probability that at most 3 men entered thedrugstore, given that 10 women entered in that hour. What assumptions have youmade? The probability is 0.265 . (Round to four decimal places asneeded.) What assumptions have youmade? Select all that apply. A. The number of men that enter the store is dependent on the number of women that enter the store. B. It is equally likely for men and women to enter the store. C. The number of men that enter the store is independent from the number of women that enter the store. D. It is not equally likely for men and women to enter the store. 4. The expected number of typographical errors on a page of a certain magazine is 0.3. What is the probability that an article of 5 pages contains (a) 0 and (b) 2 or more typographicalerrors? Explain your reasoning. Assume that the number of typographical errors on different pages are independent. The number of typographical errors on a page should approximately bea(n) Poisson Poisson random variable and the sum of independent random variables of this distribution isa(n) random variable. ( a) The probability that an article of 5 pages contains 0 typographical errors is (Round to four decimal places asneeded.) 0.2231 (b) The probability that an article of 5 pages contains 2 or more typographical errors is . 0.4422 . (Round to four decimal places asneeded.) https://tdx.acs.pearsonprd.tech/api/v1/print/highered 3/4 11/19/23, 11:37 PM HW06-Nyi Nyi Moe Htet 5. The joint probability mass function of X and Y is given by the following equations. p(1,1) = 1 4 p(1,2) = 1 8 p(2,1) = 1 8 p(2,2) = 1 2 ( a) Compute the conditional mass function of X given Y = i, i = 1, 2. (b) Are X and Yindependent? X (c) Compute P{XY ≤ 3}, P{X + Y > 2}, P >1 . Y (a) The conditional mass function of X given Y is pX|Y (1|1) = pX|Y (2|2) = 4 2 , pX|Y (2|1) = 3 1 3 , pX|Y (1|2) = 1 5 , and . 5 (Type integers or simplifiedfractions.) ( b) Select the correct choiceand, ifnecessary, fill in the answer box to complete your choice. (Type an integer or a simplifiedfraction.) A. X and Y are independent because p(1,1) ≠ pX (1)pY (1) = B. X and Y are not independent because p(1,1) ≠ pX (1)pY (1) = C. X and Y are not independent because p(1,1) = pX (1)pY (1) = . 9 64 . . D. X and Y are independent because p(x,y) = pX (x)pY (y) for all(x,y). 1 (c) P{XY ≤ 3} = 2 (Type an integer or a simplifiedfraction.) 3 P{X + Y > 2} = 4 (Type an integer or a simplifiedfraction.) P X Y >1 = 1 8 (Type an integer or a simplifiedfraction.) https://tdx.acs.pearsonprd.tech/api/v1/print/highered 4/4