need help in these problems

\f\fSuppose a certain type of small data processing firm is so specialized that some have difficulty making a profit in their first year of operation. The probability density function that characterizes the proportion Y that make a profit is given by f(y) ky'(1 - y). OSys1, 10. elsewhere. (a) What is the value of k that renders the above a valid density function? (b) Find the probability that at most 50% of the firms make a profit in the first year. (c) Find the probability that at least 80% of the firms make a profit in the first year.Magnetron tubes are produced on an automated assembly line. A sampling plan is used periodically to assess quality of the lengths of the tubes. This measurement is subject to uncertainty. It is thought that the probability that a random tube meets length specification is 0.99. A sampling plan is used in which the lengths of 5 random tubes are measured. (a) Show that the probability function of Y , the number out of 5 that meet length specification, is given by the following discrete probability function: 5! f(y) y!(5 - y)! (0.99)" (0.01)5-, (b) Suppose random selections are made off the line and 3 are outside specifications. Use fly) above either to support or to refute the conjecture that the probability is 0.99 that a single tube meets specifications.On a laboratory assignment, if the equipment is working, the density function of the observed outcome, X, is (2(1-3), 0