a. All manufacturing devices and machines fail to work sooner or later. Suppose that the failure rate is constant and the time to failure (in hours) is an exponential r.v. X with parameter . i. Measurements show that the probability that the time to failure for computer memory chips in a given class exceeds 104 hours is e-1 ( 0.368). Calculate the value of the parameter . ii. Using the value of the parameter determined in part (i), calculate the time x0 such that the probability that the time to failure is less than x0 is 0.05.

b. Intelligence Quotient (IQ) scores are normally distributed with a mean of 100 and a standard deviation of 15. i. To qualify for membership of Mensa, the organization for the "intellectually very superior", one requires an IQ score above 131.5. What percentage of people meet the Mensa eligibility requirement? ii. If we redefine the category of 'intellectually very superior" to be scores in the top 2.28 %, what does the minimum score now become for Mensa membership eligibility? iii. Calculate the quartiles of the distribution.