IE 242: Probability and Statistics Homework 2: Due Monday 30/3/2015 @ 2:00pm Problem 1: Let the random variable X have a discrete uniform distribution on the integers 0 ≤ x ≤ 99. Determine the mean and variance of X . Problem 2: A particularly long traffic light on your morning com- mute is green 20% of the time that you approach it. Assume that each morning represents an independent trial. (a) Over 5 mornings, what is the probability that the light is green on exactly one day? (b) Over 20 mornings, what is the probability that the light is green on exactly four days? (c) Over 20 mornings, what is the probability that the light is green on more than four days? Problem 3: Samples of 20 parts from a metal punching process are selected every hour. Typically, 1% of the parts require rework. Let X denote the number of parts in the sample of 20 that require rework. A process problem is suspected if X exceeds its mean by more than 3 standard deviations. (a) If the percentage of parts that require rework remains at 1%, what is the probability that X exceeds its mean by more than 3 standard deviations? (b) If the rework percentage increases to 4%, what is the probability that X exceeds 1? (c) If the rework percentage increases to 4%, what is the probability that X exceeds 1 in at least one of the next five hours of samples? Problem 4: Printed circuit cards are placed in a functional test after being populated with semiconductor chips. A lot contains 140 cards, and 20 are selected without replacement for functional testing. (a) If 20 cards are defective, what is the probability that at least 1 defective card is in the sample? (b) If 5 cards are defective, what is the probability that at least 1 defective card appears in the sample? Problem 5: The number of flaws in bolts of cloth in textile manufacturing is assumed to be Poisson distributed with a mean of 0.1 flaw per square meter. (a) What is the probability that there are two flaws in one square meter of cloth? (b) What is the probability that there is one flaw in 10 square meters of cloth? (c) What is the probability that there are no flaws in 20 square meters of cloth? 1 Prepared by Dr. Hussam Alshraideh (d) What is the probability that there are at least two flaws in 10 square meters of cloth? Problem 6: The probability density function of the length of a metal rod is f (x) = 2 for 2.3 < x < 2.8 meters. (a) If the specifications for this process are from 2.25 to 2.75 meters, what proportion of rods fail to meet the specifications? (b) Assume that the probability density function is f (x)= 2 for an interval of length 0.5 meters. Over what value should the density be centered to achieve the greatest proportion of rods within specifications? Problem 7: The diameter of a particle of contamination (in micrometers) is modeled with the 2 probability density function f (x) = 3 for x > 1. Determine the following: x (a) P (X < 2) (b) P (X > 5) (c) P (4 < X < 8) (d) P (X < 4 or X > 8) (e) x such that P (X < x) = 0.95 Problem 8: The time until recharge for a battery in a laptop computer under common conditions is normally distributed with a mean of 260 minutes and a standard deviation of 50 minutes. (a) What is the probability that a battery lasts more than four hours? (b) What are the quartiles(the 25% and 75% values) of battery life? (c) What value of life in minutes is exceeded with 95% probability? Problem 9: The fill volume of an automated filling machine used for filling cans of carbonated beverage is normally distributed with a mean of 12.4 fluid ounces and a standard deviation of 0.1 fluid ounce. (a) What is the probability that a fill volume is less than 12 fluid ounces? (b) If all cans less than 12.1 or more than 12.6 ounces are scrapped, what proportion of cans is scrapped? (c) Determine specifications that are symmetric about the mean that include 99% of all cans. 2 Prepared by Dr. Hussam Alshraideh Problem 10: The life of a semiconductor laser at a constant power is normally distributed with a mean of 7000 hours and a standard deviation of 600 hours. (a) What is the probability that a laser fails before 5000 hours? (b) What is the life in hours that 95% of the lasers exceed? (c) If three lasers are used in a product and they are assumed to fail independently, what is the probability that all three are still operating after 7000 hours? Problem 11: The life of automobile voltage regulators has an exponential distribution with a mean life of six years. You purchase a six-year-old automobile, with a working voltage regulator and plan to own it for six years. (a) What is the probability that the voltage regulator fails during your ownership? (b) If your regulator fails after you own the automobile three years and it is replaced, what is the mean time until the next failure? Problem 12: The length of time (in seconds) that a user views a page on a Web site before moving to another page is a lognormal random variable with parameters θ = 0.5 and ω 2 = 1. (a) What is the probability that a page is viewed for more than 10 seconds? (b) By what length of time have 50% of the users moved to another page? (c) What are the mean and standard deviation of the time until a user moves from the page? 3 Prepared by Dr. Hussam Alshraideh