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1 Graded Homework Continued #4 Due 3/31 1. Daily sales records for a computer-manufacturing firm show that it will sell 0, 1 or mainframe computer systems manufactured at an eastern plant with probabilities as listed. Number of Sales(N 1 ) 0 1 Probability(P(N 1 = n 1 ) A. What is the cumulative distribution function of the random variable N 1? B. What are the mean and the variance of N 1? C. Suppose each computer system costs $00,000. What are the mean and the standard deviation of the daily revenue R? D. Suppose that for a western plant the corresponding probability distribution is given by Number of Sales(N ) 0 1 Probability(P(N =n ) Assuming the number of sales in the east and the west are independent random variables what is the probability distribution of the total number of sales T = N 1 + N?. Do the following problem about the binomial distribution using computer software. Use either Mathematica or Minitab. Seventy-four college students are interviewed about the quality of food in the college cafeteria. The probability that a student will reply that the quality of food is satisfactory is A. What is the probability that exactly 30 students will reply that the quality of the food is satisfactory? B. What is the probability that at least 35 students will reply that the quality of the food is satisfactory? C. What is the probability that between 30 and 35 students inclusive will reply that the quality of the food is unsatisfactory?

2 #5 Due 4/7 1. A lot of 80 washers contains five in which the variability in thickness about the circumference of the washers is unacceptable. A sample of washers is selected at random without replacement. A. What is the probability that : (1) at most one of the washers is unacceptable? () at least two unacceptable washers is in the sample? (3) exactly two unacceptable washers is in the sample? B. What is the mean and the variance of (1) the number of unacceptable washers? () the number of acceptable washers?. The probability that your call to a service line is answered in less than 30 seconds is Assume your calls are independent. A. If you call 15 times what is the probability that exactly 8 of your calls are answered within 30 seconds? B. What is the probability that you must call exactly four times to obtain the first answer within 30 seconds? C. What is the probability that you must call at least four times to obtain the first answer within 30 seconds? D. What is the probability that you must call exactly times for three of your calls to be answered within 30 seconds? Use computer Software to do problem The probability that a computer chip is defective is.00. Find the probability that at most 15 of the next 000 chips are defective (1) Using the binomial distribution. () Using the Poisson approximation to the binomial distribution.

3 #6 Due 4/14 1. Consider the density function f(x) = ke.(x 5) x > 5 0 elsewhere 1. Find k so that f(x) is a bonafide probability density function. (You may do the integrals by hand, calculator or computer algebra provided that you indicate clearly which integrals are being evaluated and what tools are being used. B. Find the exact value of P(µ 1.5σ X µ+1.5σ). C. Find the lower bound of the probability in B using Chebychev s Theorem. How does it compare with the exact value?. Given the probability density function f(x) = x < x < 5 Find A. The CDF. B. P( X 3.5) C. The mean of the random variable X. #7 Due 4/1 1.The compressive strength of samples of cement can be modeled by a normal distribution with a mean of 6000 kilograms per square centimeter and a standard deviation of 0 kilograms per square centimeter. A. What is the probability that a sample s strength is less than 650 Kg/cm? B. What is the probability that a sample s strength is between 5800 and 5900 Kg/cm?.Do this problem using the software two ways. First use the normal approximation to the binomial distribution with the continuity correction. Then calculate the exact binomial probabilities. Compare the results. The manufacturing of semiconductor chips produces % defective chips.assume that the chips are independent and that a lot contains 00 chips. A. Appoximate the probability that more than 0 but at most 30 chips are defective? B. Approximate the probability that exactly 0 chips are defective

4 #8 Due 4/8 1. Use Computer software when appropriate The time between dialups to an internet provider is an exponential random variable with a mean of 4 minutes. A. Suppose a customer dialed in 1 minutes ago. What is the probability another customer arrives during the next 0.5 minutes? B. What is the probability that more than four customers call in the next minutes? C. What is the mean and the variance of the time until 5 customers dial in?. The life of a recirculating pump follows a Weibull distribution with parameters α = /3 and β = 00 hours. A. Determine the mean and the variance of the life of the pump. B. What is the probability the pump will fail to last 500 hours? C. What is the probability the pump will last longer than its mean? D. What is the probability the pump will last longer than its median? #9 Due 5/5 1. Suppose discrete random variables X and Y have the joint probability mass function y / x Find A. P[X + Y < 1] B. Whether X and Y are independent.. Find the marginal probability mass functions and all of the conditional probability mass functions. E. Find the correlation coefficient. F.Find Var(X Y)

5 . Suppose continuous random variables X and Y have joint distribution f(x,y)= kx y 3 0 < y < x < 1 0 elsewhere A. Find k so that f(x,y) is a bonafide probability distribution. 1. Find the marginal densities g(x) and h(y).. Find P(X<0., Y<0.1] D. Find P[Y<1-X] E. Find f(x 1 ) 3 F. Find P(X < 1 Y = 1 ) Suppose discrete random variables X and Y have the joint probability mass function y \ x Show that X and Y is uncorrelated but not independent. 4. Let continuous random variables X and Y have joint density f(x,y)= 1x3 y 0 < y < x < 1 0 elsewhere Find cov(x,y)

6 # Due 5/1 1. Use Minitab to do this problem. Please hand in the computer printout. [Hint: Put all of the data in column C1. Then use the commands stem and leaf c1; increment =. describe c1 box plot c1 The joint temperatures of the O rings (degrees F) for each test firing or actual launch of the space shuttle rocket motor are A. Make a stem and leaf display. B. Make a five number summary. C. Make a box plot. D. Find the mean and the standard deviation. E. Does the data appear to be normally distributed skewed symmetric? Explain. F.Are there any outliers? 3. The number of accidents at a hazardous intersection per week varies with mean. and standard deviation 1.4. Let x be the mean number of accidents at the intersection in 0 weeks. A.Find P(x <.1) B.What is the approximate probability that there are fewer than 05 accidents during 0 weeks? C. What is the 90 th percentile of x?