PSTAT 120B, Spring 2017 Homework 4 Due May 15, 2017 1. Suppose we compare three estimators, 1, 2 and 3 , for . We know that E( 1 ) = E 2 = Var( 1 ) = 12 Var( 2 ) = 22 . If 3 = a 1 + (1 a) 2, (a) Show that 3 is an unbiased estimator for . (b) Show how to choose a so that variance of 3 is minimized. Assume 1 and 2 are independent. 2. A car manufacturer numbers their tanks uniformly over the interval (1, ). Suppose X1 , X2 , . . . , Xn denotes a random sample of tanks. is a biased estimator of . (a) Compute the bias and show that X (b) Find a function of X that is an unbiased estimator of . is used as an estimator of . (c) Find the MSE when X 3. Suppose Y1 , Y2 , Y3 denote a random sample from an exponential distribution with density function ( y/ e , y >0 f (y) = 0, otherwise Consider the following estimators: 1 = Y1 Y1 + Y2 2 = 2 Y + 2Y2 1 3 = 2 4 = min(Y1 , Y2 , Y3 ) 5 = Y (a) Which estimators are unbiased? (b) Among the unbiased estimators, which has the smallest variance? 4. Let Y1 , Y2 , . . . , Yn denote a random sample of size n from a population whose density is given by ( 1 y , 0 y , f (y) = 0, elsewhere where > 0 is a known, fixed value, but is unknown. Consider the estimator = max(Y1 , Y2 , . . . , Yn ). (a) Show that is a biased estimator for . (b) Find a multiple of that is an unbiased estimator of . (c) Derive MSE(). 5. A survey is used to quantify the support for an increase in education spending. The pollster tells us that the approval rating for the increase is 60% with 4% standard error. Respondents have been modeled with independent identically distributed Bernoulli random variables. 1 PSTAT 120B, Spring 2017 Homework 4 Due May 15, 2017 (a) Approximately how many people did the pollster survey? (b) Even if the true approval rating is 2 standard error away from the pollster's estimate, there would be majority support for the increased spending. What is the lower bound on the probability that approval rating is within 2 standard errors. Hint: use Chebyshev's inequality. 6. The number of persons coming through a blood bank until the first person with type A blood is found is a random variable Y with a geometric distribution. If p denotes the probability that any one randomly selected person will possess type A blood, then E(Y ) = 1/p and Var(Y ) = (1 p)/p2 . (a) Find a function of Y that is an unbiased estimator of V (Y ). (b) Suggest how to form a 2-standard-error bound on the error of estimation when Y is used to estimate 1/p. 7. The reading on a voltage meter connected to a test circuit is uniformly distributed over the interval (, + 1), where is the true but unknown voltage of the circuit. Suppose that Y1 , Y2 , . . . , Yn denote a random sample of such readings. (a) Show that Y is a biased estimator of and compute the bias. (b) Find a function of Y that is an unbiased estimator of . (c) Find MSE(Y ) when Y is used as an estimator of . 8. The number of breakdowns per week for an electronic gadget is a random variable Y with a Poisson distribution and mean . A random sample Y1 , Y2 , . . . , Yn of observations on the weekly number of breakdowns is available. (a) Suggest an unbiased estimator for , based on Y1 , Y2 , . . . , Yn . (b) The weekly cost of repairing these breakdowns is C = 3Y + Y 2 , if Y is the number of breakdowns that week. Show that E(C) = 4 + 2 . (c) Find a function of Y1 , Y2 , . . . , Yn that is an unbiased estimator of E(C). Hint: Use what you know about E(Y ) and E(Y 2 ). 2