ECON 3130 SECTION 3Christopher Rojas Yu SheDiscrete Random Variable & Moment Generating Function1 / 6 September 8, 2017 1 / 6IntroductionWe will work on 4 exercises today.2 / 6 September 8, 2017 2 / 6Moment Generating FunctionIf the moment generating function of X isM(t) =14et+18e2t+38e3t+14e4tfind the mean, variance and pmf of X .3 / 6 September 8, 2017 3 / 6Binomial DistributionExercise 2.4-4It is claimed that 15% of the ducks in a particular region have patent schis-tosome infection. Suppose that seven ducks are selected at random. Let Xequal the number of ducks that are infected.(a) Assuming independence, how is X distributed?(b) Give the values of the mean and variance of X(c) Find the following probabilities: (i) P(X ≥ 2), (ii) P(X = 1), (3)P(X ≤ 3)4 / 6 September 8, 2017 4 / 6Binomial DistributionExercise 2.4-15A hospital obtains 40% of its flu vaccine from Company A, 50% from Com-pany B, and 10% from Company C. From past experience, it is known that3% of vials from A are ineffective, 2% from B are ineffective, and 5% fromC are ineffective. The hospital tests five vials from each shipment. If atleast one of the five is ineffective, find the conditional probability of thatshipment’s having come from C.5 / 6 September 8, 2017 5 / 6Binomial DistributionExercise 2.4-18In group testing for certain disease, a blood sample was taken from each of nindividuals and part of each sample was placed in a common pool. The latterwas then tested. If the result was negative, there was no more testing and alln individuals were declared negative with one test. If, however, the combinedresult was found positive, all individuals were tested, requiring n + 1 tests.If p = 0.05 is the probability of a person’s having the disease and n = 5,compute the expected number of test needed, assuming independence.6 / 6 September 8, 2017 6 /