Mathematics for Business Decisions, Part IIHomework 6Mathematics for Business Decisions, Part IIHomework 6forMath 115b, Section: Instructor: James BarrettDate: byTeam: We, the undersigned, maintain that each of us participated fully and equally in the completion of this assignment and that the work contained herein is original. Furthermore, we acknowledge that sanctions will be imposed jointly if any part of this work is found to violate the Student Code of Conduct, the Code of Academic Integrity, or the policies and procedures established for this course. Name Typed Signature % of Grade__________________ ______________________________ ______________________________ ______________________________ ______________________________ ______________________________ ______________________________ _____________________________ ____________1. Let X be the random variable whose c.d.f. is given below.FX( x )={0 .0 if x<20 .2 if 2≤x <40 .5 if 4≤x <60 .7 if 6≤x<81. 0 if 8≤x(i) Find the p.m.f for X.(ii) Find the following probabilities:P(2≤X ≤6)P( X=8)P( X≥4 )P( X=5)P( X >7 )P( X≤2)(iii) Compute the E(X).Solution:2. Let X be a binomial random variable with n=20 and p=0 . 65. Use the Binomial Distribution function in Excel to compute (i) fX(14 ) and (ii) FX(14 ). (iii) Plot a graph of the p.m.f., fX( x ). (iv) Plot a graph of the c.d.f., FX( x). Your plots should resemble those made in Graphs of Binomial 2.xls.Solution:3. Consolidated Widgets Inc. has a force of 130 salespeople, whose sales performances can be considered to be independent of each other. The manager estimates that the probability of any one sales person reaching his or her monthly quota is 0.6. (i) What is the probability that at least 70 people make their quota? (ii) What is the probability that at most 70 people make their quota?Solution:4. Use Experimentation in Binomial 2.xlsx and the situation in Example 2 to find the smallest value of p, rounded to 2 decimal places, that the saleswoman would need in order for the probability of at least 7 sales per day to be greater than or equal to 0.6. Assume that she makes 32 contacts dailySolution5. Refer to Problem 3. Let Q be the number of salespeople who make their quotasin a given month. (i) Use the special formula for the expected value of a binomial random variable to compute E(Q). (ii) In the real world language of people andsales, what is the interpretation of the expected value of Q?Solution:6. Use Integrating.xlsm to determine whether or notf (x )={−x2+2⋅x if 0≤x≤20 elsewhere could be a p.d.f. for some continuous random variable. Solution:7. Use the p.d.f. and c.d.f. defined in Example 5. Compute the probability that the time between computer system shutdowns will be at least 24 hours. (i) Do this with the p.d.f, using Integrating.xlsm (ii) Do this with the c.d.f.,.Solution:8. Let X be an exponential random variable with parameter α=0 .5. (i) UseFX to compute P(0 .2≤X≤1. 2). (ii) Use fX to compute the same probability.Solution:9. The lifetime, in years, of the transmitter in your new wireless mouse is an exponential random variable with parameter α=3 . 8 years. What is the probability that the transmitter lasts for at least 4 and 1/2 years.Solution:10. Let X be a continuous random variable that is uniform on the interval [0, 10 ]. (i) What is the probability the X is at most 6.87? (ii) What is the probability that X is no less than 5.59?Solution:11. Let W be the working lifetime, measured in years, of the microchip in your new digital watch. The manufacturer claims that W has an exponential distribution with parameter α=5 years. (i) Assuming that this is correct, use Integrating.xlsm and the probability density function fW to verify that the expected value of W is 5 years. Compute the probabilities that the chip lasts for (ii)at least 9 years and (iii) at most 4 years. Do two way using f(x) and Integrating .xlsm and by using F(x).Solution:12. Let W be the random variable described in Problem 11. Suppose that the actual mean of W is 3-1/2 years. Use Integrating.xlsm and the probability density function fW to compute the probabilities that the chip lasts for (i) at least 9 yearsand (ii) at most 4 years. Also answer each question using F(x).Solution:13. Let B be the random variable giving the number of checks that your business takes in per month that are returned for insufficient funds. Your records for the last8 months show 14, 19, 8, 8, 21, 12, 17, and 6 such checks. (i) Use this informationto approximate the expected value of B. (ii) What is the real world interpretation of your answer to Part (i)? (iii) Is your answer to Part (i) the exact value ofE( B)? Explain your answer.Solution. 14. Let S be the continuous random variable giving your company’s weekly gross sales in thousands of dollars. Records contain the sample values of S that are givenin the sheet Exercise Data 2 of Samples.xlsx. (i) Compute the minimum, maximum, and mean of the sample. (ii) Select suitable bin limits and use the HISTOGRAM function to plot an approximation of the p.d.f., fS. (iii) Do you think that it is very likely that S is an exponential random variable?Solution:15. Use Excel and your team’s data for the signals on past oil leases to compute the complete set of errors in the signals. Find the sample mean of these errors and plot a histogram which approximates the actual p.d.f., fR, of