What is the solutions for the questions below?

Ex. 4.72. A manufacturing company has developed a machine for cleaning carpet that is fuel-efcient because it delivers carpet cleaner so rapidly. Of interest is a random variable y, the amount in gallons per minute delivered. It is known that the density function is given by _ 1, 7 S y S 8; f(y) _ { 0, elsewhere. (a) Sketch the density function. (b) Give E(Y), E(Y2), and Var(Y). [Answerz (b) 15/2; 169/3; 1/12.] Ex. 4.76. Suppose 120 new jobs are opening up at an automobile manufacturing plant and 1000 applicants show up for the ~70 120 positions. To select the best 120 from among the applicants, the company gives a test that covers mechanical skill, manual dexterity, and mathematical ability. The mean grade on this test turns out to be 60, and the scores have a standard deviation of 6. Can a person who scores 78 count on getting one of the jobs? @: Use Chebyshev's theorem.] Assume that the distribution is symmetric about the mean. Ex. 4.78. Compute Pm 20 2; 9(56) _ { 8, elsewhere. and _ 2y, 0