(a) Let X1,X2, ,Xn be a random sample from a distribution with pdf f(x;), where is an unknown parameter. T(X1,X2, ,Xn) is a function of X1,X2, ,Xn used to estimate . Explain the meaning of: (i) T(X1,X2, ,Xn) is an unbiased estimator of . [2 marks] (ii) T(X1,X2, ,Xn) is a consistent estimator of . [2 marks] (b) The amount of time, Y , in minutes that a bank teller spends with a customer is known to have an exponential distribution with an average amount of 7 minutes. The density function is dened by f(y) = 1 7e1 7y, y > 0 0, elsewhere Find the probability that the bank teller spends seven to nine minutes with a randomly selected customer. [4 marks]

(c) The time taken to assemble a car in a certain plant is a random variable having a normal distribution with a mean of 20 hours and a standard deviation of 2 hours. What is the probability that a car can be assembled at this plant in a period of (i) less than 19.5 hours [2 marks] (ii) between 20 hours and 22 hours [3 marks]

(d) A continuous random variable X has a probability density function given by f(x) =(1 4x, 0 x 2 0, otherwise Obtain: (i) the CDF of X [3 marks] (ii) P(0.5 X 2) [2 marks] (e) A laptop manufacturing company claims that its top quality laptops are good, on average, for atleast 200 months before expiry. A consumer protection agency tested 60 such laptops and found that on average they last 265 months with a standard deviation of 10 months. Construct: (i) 90% condence interval for the mean. [3 marks] (ii) 99% condence interval for the mean [3 marks] (iii) Which condence interval should the agency use for better precision? Why? [2 marks]

(f) A Mathematical Statistician determines that the revenue a City Restaurant makes in a week is a random variable, X, with moment generating function given by

MX(t) =

1 (12500t)4 Calculate the variance the of the revenue the city restaurant makes in a week. [4 marks