Homework 5: Distributions And Random Variables

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

2023
Views:

232

Pages:

6
Author:

Nyi Moe

11/19/23, 11:37 PM HW05-Nyi Nyi Moe Htet Student: Nyi Nyi Moe Htet Date: 11/19/23 Instructor: Xiaolong Han Course: CSUN_MATH340_HAN (1) Assignment: HW05 1. The probability density function ofX, the lifetime of a certain type of electronic device(measured inhours), is given by the following equation. Complete parts(a) through(c). 5 f(x) = x>5 2 x 0 x≤5 (a) Find P{X > 15}. 1 P{X > 15} = 3 (Simplify youranswer.) (b) What is the cumulative distribution function ofX? 1− F(x) = 5 x 0 x>5 x≤5 (c) What is the probability that of 6 such types ofdevices, at least 3 will function for at least 10 hours? The probability is 21 32 . (Simplify youranswer.) What assumptions are youmaking? A. The events that each device exceeds 10 hours are independent. B. The events that each device exceeds 10 hours are exponentially distributed. C. The events that each device exceeds 10 hours are normally distributed. D. The events that each device exceeds 10 hours are dependent. 2. Trains headed for a destination A arrive at the train station at 15-minute intervals starting at 8 A.M., whereas trains headed for destination B arrive at 15-minute intervals starting at 8:05 A.M. Complete parts(a) and(b). ( a) If a certain passenger arrives at the station at a time uniformly distributed between 8 and 10 A.M.(noninclusive) and then gets on the first train thatarrives, what proportion of time does he or she go to destinationA? 2 3 (Simplify youranswer.) (b) What if the passenger arrived at a time uniformly distributed between 8:10 and 10:10 A.M(noninclusive)? 2 3 (Simplify youranswer.) https://tdx.acs.pearsonprd.tech/api/v1/print/highered 1/6 11/19/23, 11:37 PM HW05-Nyi Nyi Moe Htet 3. You arrive at a bus stop at 10A.M., knowing that the bus will arrive at some time uniformly distributed between 10 and10:30. ( a) What is the probability that you will have to wait more than 18 minutes? (b)If, at10:15, the bus has not yetarrived, what is the probability that you will have to wait at least an additional 6 minutes? (a) The probability is 2 5 . (Simplify youranswer.) (b) The probability is 3 5 . (Simplify youranswer.) https://tdx.acs.pearsonprd.tech/api/v1/print/highered 2/6 11/19/23, 11:37 PM HW05-Nyi Nyi Moe Htet 4. If X is a normal random variable with parameters μ = 10 and σ2 = 36, compute the following. ( a)P{X > 6} (b)P{4 < X < 16} (c)P{X < 9} (d)P{X < 13} (e)P{X > 16} Click here to view the standard normal distribution table.1 ( a)P{X > 6} = 0.7486 (Round to four decimal places asneeded.) ( b)P{4 < X < 16} = 0.6826 (Round to four decimal places asneeded.) ( c)P{X < 9} = 0.4325 (Round to four decimal places asneeded.) ( d)P{X < 13} = 0.6915 (Round to four decimal places asneeded.) ( e)P{X > 16} = 0.1587 (Round to four decimal places asneeded.) 1: Standard Normal Distribution Table https://tdx.acs.pearsonprd.tech/api/v1/print/highered 3/6 11/19/23, 11:37 PM HW05-Nyi Nyi Moe Htet 5. Let X be a normal random variable with a mean of 14 variance of 16. Find the value of c such that P{X > c} = 0.20. Click here to view the standard normal distribution table.2 c= 17.36 (Round to two decimal places asneeded.) 2: Standard Normal Distribution Table https://tdx.acs.pearsonprd.tech/api/v1/print/highered 4/6 11/19/23, 11:37 PM HW05-Nyi Nyi Moe Htet 6. Suppose that theheight, ininches, of a25-year-old man is a normal random variable with parameters μ = 69 and σ2 = 6.76. What percentage of25-year-old men are taller than 6feet, 2inches? What percentage of men in the6-footer club are taller than 6 feet, 5inches? 3 Click the icon to view the standard normal distribution table. Approximately 2.74 % of25-year-old men are taller than 6feet, 2 inches. (Round to two decimal places asneeded.) Approximately 0.80 % of25-year-old men in the6-footer club are taller than 6feet, 5 inches. (Round to two decimal places asneeded.) 3: Standard Normal Distribution Table 7. 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 5/6 11/19/23, 11:37 PM 8. HW05-Nyi Nyi Moe Htet The time(in hours) required to repair a machine is an exponentially distributed random variable with parameter λ = 1 . What is 3 ( a) the probability that a repair time exceeds 2 hours? (b) the conditional probability that a repair takes at least 12 hours, given that its duration exceeds 10 hours? − (a) The probability that a repair time exceeds 2 hours is e 2 3 . (Type an exact answer in simplified form. Use integers or fractions for any numbers in theexpression.) − (b) The conditional probability that a repair takes at least 12 hours, given that its duration exceeds 10 hours is e 2 3 . (Type an exact answer in simplified form. Use integers or fractions for any numbers in theexpression.) 9. If X is uniformly distributed over ( − 1,1), find the following. (a) P X > (a) P X > 1 ; 2 (b) the density function of the random variable X . 1 1 = 2 2 (Simplify youranswer.) (b) Let A = X . f(a) = 1 , 0 0 , otherwise 0, show that cX is exponential with parameter The probability that cX is less than x is given by P (Type aninequality.) cX < x Dividing both sides of this inequality by c yields P X< x c λ c . . , assuming c > 0. (Type aninequality.) If X is an exponential random variable with parameter λ, then the probability in the previous step is equal to that cX is exponential with parameter λ c 1−e − λ x c . This shows . (Type an exact answer in terms of e.) https://tdx.acs.pearsonprd.tech/api/v1/print/highered 6/6