0.1 Exercises for Chapter 31. An unfair die with six sides is rolled. Let X be outcome of the die. The probabilitymass 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 Is legitimate Is not legitimatep1(x) .2 .3 .4 .2p2(x) .3 .3 .5 -.1p3(x) .1 .4 .4 .1b) Calculate E(X) and E(1/X) using the following pmf:x 1 2 3 4p(x) .4 .3 .1 .2c) In a win-win game, the player will win a monetary price, but he/she has to decidebetween a fixed and a random price. In particular the player is offered$1000E(X)or$1000X,where the random variable X has the distribution given in (b). When the choiceis made a value of X is generated and the player receives the chosen price. Whichchoice would you recommend the player to make?3. A metal fabricating plant currently has five major pieces under contract each witha deadline for completion. Let X be the number of pieces completed by theirdeadlines. Suppose that X is a random variable with p.m.f. p(x) given byx 0 1 2 3 4 5p(x) .05 .10 .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 cumulativedistribution function of the random variable X isF (x) =0 x < 00.2 0 ≤ x < 10.7 1 ≤ x < 20.9 2 ≤ x < 31 3 ≤ x(a) Plot the cumulative distribution function. What is the probability of two ormore 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 isthe expected revenue per day in dollars. Find the mean value and variance ofY .5. The cumulative (probability) distribution function of checkout duration in a certainsupermarket isF (x) =x24, for x between 0 and 2 .(Clearly F (x) = 0 for x ≤ 0 and F (x) = 1 for x > 2.)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 reserveat McAllister is checked out by a randomly selected student, and suppose that Xhas density functionf(x) =(0.5x 0 ≤ x ≤ 20 otherwiseWhat is the probability that the book is checked out between 0.5 and 1.5 hours?27. Let X denote the resistance of a randomly chosen resistor, and suppose that its pdfis given byf(x) =(kx if 8 ≤ x ≤ 100 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, thatis,f( x) = e−xwhere 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, iswidely used commercially to grow the stem of one variety that produces fine fruiton the root system of another variety with a hardy root system. For example, mostsweet oranges grow on trees grafted to the root of a sour orange variety. Supposethat each graft fails independently with probability 0.3. Five grafts are scheduledto be performed next week. Let X denote the number of grafts that will fail nextweek.(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.3iii. The variance of the cost from the failed grafts.10. In the grafting context of the previous exercise, suppose that grafts are done one ata 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 theunderside 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 havecracks.(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 offive ipods and thoroughly inspects the click wheel of each of them. Suppose thatthe new shipment of 20 ipods contains three with malfunctioning click wheel. LetX 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 andindependently 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 withparameter λ is F (x) = 1 − exp(−λx). Give an expression (involving λ) for theprobability P (2 < X < 5).(b) Let now Y denote the same life time but in minutes instead of hours. Give anexpression for the cumulative distribution function of Y . (Hint: Y ≤ y holdsif 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 madefixture is a random variable with pdff(x) =(λe−λ(x−θ)if x ≥ θ0 otherwise,with λ = 0.02, θ = 48. An architect overseeing a renovation must order a custommade piece to replace an old fixture which unexpectedly broke. The architect hasdetermined that it will cost him 200 dollars a day for every day beyond three daysthat he does not have the piece.(a) What is the probability that, if he orders the piece from that manufacturer, hewill lose no money?(b) What is the probability that he will lose no more that 400 dollars?(c) What is the probability that he will lose between 400 and 800