Department of Mathematics & Statistics STAT 2593 Midterm ...

A machine has a component which is subject to deterioration over time. A backup component issupplied, along with the machine, at the time that the machine is purchased. Suppose that theoriginal component has a lifetime (in years) which is Gamma distributed with parameters α = 30and β = 0.2; the backup component has a a lifetime (in years) which is Gamma distributed withparameters α = 10 and β = 0.5.31. How long do you expect the machine to last, all told? (I.e. what is the expected total lifetimeof the two components, original and backup)?11 years32. Assume that the lifetime of the backup component is independent of the lifetime of the originalcomponent. What is the variance of the time that the machine lasts? (I.e. what is the varianceof the total lifetime of the two components, original and backup)?3.7 years 2The breaking strength of rivets of a certain type has a mean value of 10,000 psi and a standarddeviation of 500 psi. A random sample of 40 such rivets is obtained.33. What (approximately) is the probability that the sample mean breaking strength is between9900 and 10200 psi?0.890534. What is the name of the theorem that allows you to make the foregoing approximation?The Central Limit TheoremThe rain in Spain has an annual mean of 17 inches, with a standard deviation of 4 inches. The rainin Calcutta 1 has an annual mean of 64 inches with a standard deviation of 12 inches. Assume thesetwo random variables to be normally distributed and to be independent of each other (a dubiousassumption, but make it anyway).35. What is the probability that the rainfall in Calcutta in a given year exceeds the rainfall inSpain in the same year by more than 55 inches?0.26431 The rain in Spain stays mainly in the plain; the rain in Calcutta stays mainly in the gutta.8