(8 points - 10 minutes) Mr. Johnston knew the probability of raining that morn- ing was 3/4 - he had checked the weather forecast before sleeping. The first thing he noticed when he woke up was that Ms. Johnston, who had just left home, had not taken her umbrella. Mr. Johnston knows that if it is raining, Ms. Johnston takes her umbrella with a 90% probability, but when it is not raining she does so with a 30% probability. Conditional on Ms. Johnston not taking her umbrella, what is the probability that Mr. Johnston attributed to rain?\f(10 points - 12 minutes) Let X be a Bernoulli random variable with probability function f(a) = p"(1 - p) -*, x =0,1. (a) (5 points) Derive the moment generating function (MGF) for this discrete random variable. (b) (5 points) Use the MGF obtained in (a) to derive E(X ) and Var(X).(22 points - 26 minutes) Let X1. X,1 be a random sample from me. OK) and let i]. 1;\" be a random sample from N (ply. 0%). Both samples are independent of each other. Let 0'} and of- be known and nite. W'e are interested in testing the hypothesis Ho : px = 11,- against H1 : ,ux a- y at the 1% level of signicance for a given sample. (a) (14 points) Construct a 99% condence interval for ,ux y that can be used to answer your question. You are expected to show how this interval is derived. simply stating the interval will only receive partial credit. (b) (4 points) Discuss how you would use your answer in (a) to test this hypothesis. (c) (4 points) For a particular sample. the pvalue of your test equals 0.001. What would you conclude from this result? Explain your answer. (26 points 31 minutes) A random sample X1.X2. X\" of size n is drawn from N(,u. 9). where the variance 6' is such that 0