7. Suppose th atX and Y are independent random variables that are exponential with parameter A and let X Z=. X+Y Show that for 0 1). 9. Let X1, X2 be a random sample fromN{0,1/9). Note that the sample size is 2. Find the likelihood function and use it to obtain the maximum likelihood estimator of. 10. A lie detector will show a positive reading 10% of the time when a person is telling the truth and 90% of the time when the person is lying. Suppose two people are suspects in a crime and only one is guilty. The guilty one will be lying, while the innocent one will be telling the truth . (a) What is the probability that the detector shows a positive reading ? (b) What is the probability that a person is lying when the detector shows a negative reading ? 11. In a recent survey of 500 U.S. workers, 18% say it is likely that they will lose their job in the next year. Construct a 95% confidence interval for the true proportion of U.S. workers who think they are likely to lose their job in the next year. 12. Suppose a researcher claims that the mean age of founders of start -up companies in Silicon Valley is less than 30 years. To test his claim, he randomly selected 7 startup companies and obtained the following data on the age of their founders 27, 28, 32, 33, 24, 28, 31 years. Is there significant evidence that the mean age of founders is less than 30 years? Test this at the 5% level of significance. 13. A certain electronic part is guaranteed to last at least 3 years . Seven randomly selected parts lasted for 6, 4, 6, 8, 6, 7, and 5 years. Assuming its lifetime has a uniform distribution with parameters a = 3 and some unknown b, estimate parameter b by the method of moments