0 1 Exercises for Chapter 3 1 An unfair die with six sides is rolled Let X be outcome of the die The probability mass function is given to be P X i i 21 i 1 2 6 a Show that this is a legitimate pmf b Find and plot the cumulative distribution function 2 Put a check mark in the table below indicating whether or not each of p1 x p2 x p3 x is a legitimate probability mass function pmf for the random variable X x 0 1 2 3 p1 x 2 3 4 2 p2 x 3 3 5 1 p3 x 1 4 4 Is legitimate Is not legitimate 1 b Calculate E X and E 1 X using the following pmf x 1 p x 2 3 4 4 3 1 2 c In a win win game the player will win a monetary price but he she has to decide between a fixed and a random price In particular the player is offered 1000 E X or 1000 X where the random variable X has the distribution given in b When the choice is made a value of X is generated and the player receives the chosen price Which choice would you recommend the player to make 3 A metal fabricating plant currently has five major pieces under contract each with a deadline for completion Let X be the number of pieces completed by their deadlines Suppose that X is a random variable with p m f p x given by x p x 0 1 05 10 2 3 4 5 15 25 35 10 a Find and plot the cdf of X b Use the cdf to find the probability that between one and four pieces inclusive are completed by deadline 1 c Find the expectation of X d Find the variance of X 4 Let X denote the daily sales for a computer manufacturing firm The cumulative distribution function of the random variable 0 0 2 F x 0 7 0 9 1 X is x 0 0 x 1 1 x 2 2 x 3 3 x a Plot the cumulative distribution function What is the probability of two or more sales in a day b Write down the probability mass function of X c Calculate the expected value and variance of X d Suppose the firm makes an average of 1 200 per sale Thus Y 1 200 X is the expected revenue per day in dollars Find the mean value and variance of Y 5 The cumulative probability distribution function of checkout duration in a certain supermarket is x2 for x between 0 and 2 4 Clearly F x 0 for x 0 and F x 1 for x 2 F x a Find the median checkout duration b Find the probability that the duration is between 0 5 and 1 c Find the density function f x d Find the expected value E X 6 Let X denote the amount of time for which the Stat 401 book on two hour reserve at McAllister is checked out by a randomly selected student and suppose that X has density function f x 0 5x 0 x 2 0 otherwise What is the probability that the book is checked out between 0 5 and 1 5 hours 2 7 Let X denote the resistance of a randomly chosen resistor and suppose that its pdf is given by f x kx if 8 x 10 0 otherwise a Find k b Find the mean value and variance of X c Give a formula for the cdf of X d Find P 8 6 X 9 8 8 Let X be a continuous random variable with a standard exponential density that is f x e x where x 0 Find the following probabilities a P 1 X 1 b P X 1 c P X 1 X 2 d P X 2 X 1 9 Grafting the uniting of the stem of one plant with the stem or root of another is widely used commercially to grow the stem of one variety that produces fine fruit on the root system of another variety with a hardy root system For example most sweet oranges grow on trees grafted to the root of a sour orange variety Suppose that each graft fails independently with probability 0 3 Five grafts are scheduled to be performed next week Let X denote the number of grafts that will fail next week a What is the sample space i e the set of all possible values of X b Write the formula for the pmf for X c Find the expected value of X d Find the variance of X e Suppose that the cost of each failed graft is 9 00 Find i The probability that the cost from failed grafts will exceed 20 00 ii The expected cost from failed grafts 3 iii The variance of the cost from the failed grafts 10 In the grafting context of the previous exercise suppose that grafts are done one at a time What is the expected number of successful grafts until the first failed graft 11 Suppose that 8 of the 20 buses in a particular city have developed cracks on the underside of the main frame Five buses are to be selected for thorough inspection Let X denote the number of buses among the five that are inspected that have cracks a Give the sample space of X b Find the pmf of X c Find the expected value of X d Find the variance and the standard deviation of X 12 A distributor receives a new shipment of 20 ipods He draws a random sample of five ipods and thoroughly inspects the click wheel of each of them Suppose that the new shipment of 20 ipods contains three with malfunctioning click wheel Let X denote the number of ipods with defective click wheel in the sample of five a Give the sample space of X b Find the pmf of X c Find the expected value of X d Find the variance and the standard deviation of X 13 In the context of Exercise 7 find the probability that out of four randomly and independently selected resistors exactly two have resistance between 8 6 and 9 8 14 Let X denote the life time of a component when measured in hours a The cumulative distribution function of an exponential random variable with parameter is F x 1 exp x Give an expression involving for the probability P 2 X 5 b Let now Y denote the same life time but in minutes instead of hours Give an expression for the cumulative distribution function of Y Hint Y y holds if and only if X y 60 c Find the probability density function of Y 4 d Find the expected value and variance of Y 15 The time X in hours for a certain plumbing manufacturer to deliver a custom made fixture is a random variable with pdf e x if x f x 0 otherwise with 0 02 48 An architect overseeing a …