1. Question Let X1, X2, ... be i.i.d. random variables with common variance 2. Define the sample variance as 1 n 2 sn ( X i X n )2 , n 1 i 1 where X n denotes the regular sample mean. Show that the expectation of sample variance is equal to 2. Question 2. A man has 4 keys in his key chain and he knows that one of these keys opens a particular door. a) He is trying each key in his key chain with replacement to find the key that opens the door. What is the probability that he will do more than 4 trials to find the key that opens the door? b) He is trying each key in his key chain without replacement to find the key that opens the door. Sketch an Excel spreadsheet sheet that can be used to compute the probability that he will do more than 2 trials to find the key that opens the door. Make sure to describe how you would compute a confidence interval on this probability. draw (Simply a grid, label the columns and rows and put the Excel formulas you would use to estimate the quantity of interest. You only need to provide one line of formulas.) Question 3. a) If U is a uniformly distributed random variable between 0 and 1, what is the expectation of U2 ? b) Assume that you have a full period LCG with modulus 100. You use this LCG to generate 100 random numbers. You use these random numbers to estimate the expectation E{U 2 }. That is, if U1, U2, ... are the random numbers you get from the 1 100 2 LCG, then you compute U i . What would your estimate be? Give a precise 100 i 1 mm 12m 1 .) numerical answer. (Hint: Note that 1 + 4 + 9 + ... + (m-1)2 = 6 Compare your answer with part a. Question 4. Assume that X and Y are independent and uniformly distributed over [0.1] . Compute the expectation and variance of (X+1)/(Y+2).