ECON 3130 DEPARTMENT OF ECONOMICSINTRODUCTION TO PROBABILITY AND STATISTICS CORNELL UNIVERSITYFALL 2017Problem Set #1Due: September 13, 8:40amYou may work with a study group of your choice on this assignment. How-ever, even if you work with others you must write up and submit your ownassignment. Late assignments will not be accepted.1. Consider flipping a fair (unbiased) coin five independent times.(a) What is the probability of HHHHH?(b) What is the probability of HHHTT?(c) What is the probability of 3 heads occurring in the 5 tosses?(d) What is the probability that the first three tosses are heads?(e) What is the probability of 5 heads in the 5 tosses, conditional on the firstthree tosses are heads?(f) What is the probability of 3 heads in the 5 tosses, conditional on the firstthree tosses are heads?(g) Suppose a person decides to flip a coin 5 times. After flipping the cointhree times, and discovering that the first three tosses are all heads, theperson thinks it is very unlikely to get 5 heads in a row, and thus thinksthat the next two tosses are likely to be tails to balance things out. Whatis right or wrong with this thinking?2. Prove the following assertions:(a) E ⊆ F implies E ∩ F = E(b) E ⊆ F and F ⊆ G implies E ⊆ G(c) (E ∩ F )0= E0∪ F0(d) (E ∪ F )0= E0∩ F03. Prove the following assertions:(a) A and B are independent if and only if A0and B0are independent.1(b) A and B are independent if and only if P (A|B) = P (A).(c) A ⊆ B, and P (A) 6= 0, P (B) 6= 1 then A and B are not independent.(d) If P (A) = 0, then A is independent of B for any B.(e) If P (A) = 1, then A is independent of B for any B.(f) P (A ∩ B ∩ C) = P (A|B ∩ C)P (B|C)P (C).4. In a population of 100 undergrads, 20 are seniors, 25 are juniors, and 30 arefreshmen. Given that 6 of the seniors, 5 of the sophomores, and 3 of thefreshmen smoke, and that79of the upperclassmen (juniors and seniors) don’tsmoke, find the probability that a student selected at random is(a) a smoker(b) a sophomore(c) a junior nonsmoker(d) not a senior and doesn’t smoke(e) a freshmen or a nonsmoker(f) a senior or a smoking freshman5. (HIV Testing.) According to the Centers for Disease Control and Prevention,1 in every 10,000 pre-screened volunteer blood donors in the United Statesis infected with the Human Immunodeficiency Virus, or HIV. To reduce thechance of transmitting this virus to other individuals, each unit of donatedblood is subject to an immunoassay test before the blood is used for trans-fusion. This test yields one of two possible outcomes: positive, which indi-cates that the blood donor has antibodies indicating the presence of HIV, ornegative, which indicates that the donor does not have antibodies indicatingthe presence of HIV. The standard immunoassay test is not 100% accurate.On average, the test erroneously returns a negative test result for 1 in every100 donors that are, in fact, infected with HIV. Moreover, the test erroneouslyreturns a positive test result for 1 in every 1000 donors that are, in fact, not in-fected with HIV. Disclaimer: The numbers here are somewhat dated. (Note:Because of the small probabilities involved in this problem, carry out all in-termediate calculations to at least seven decimal places or, better yet, do allcalculations in Excel or Stata.)(a) What is the probability that a unit of blood from a randomly selectedpre-screened volunteer blood donor in the U.S. will test positive?(b) A unit of blood is tested, and the test result is positive. Given this test re-sult, what is the probability that the donor does not actually have HIV?26. Consider rolling a fair die twice, and let S denote all combinations of possibleoutcomes from the two rolls,S = {(1, 1), (1, 2), ..., (1, 6), (2, 1), (2, 2), ...(6, 6)}.Suppose that any outcome in S is equally likely, P [(i, j)] = P [(i0, j0)] for all(i, j), (i0, j0) ∈ S. Let X denote the sum of the two numbers from rolling thedie twice.(a) Find P [(i, j)] for each (i, j) ∈ S.(b) Find P [X = x] for x ∈ {2, 3, ..., 12}.(c) What is P [X = x] for x 6∈ {2, 3, ..., 12}?(d) Which value of X is the most likely? in other words, find the value x∗such that P [X = x∗] ≥ P [X = x] for all x. Give an intuitive explanationfor why that value is the most likely.(e) Let F (t) = P (X ≤ t) denote the cumulative distribution function of X.Find F (t) for all t.(f) Graph F (t).(g) Let E(X) =P12x=2xP [X = x]. Find E(X).(h) How does E(X) compare to the x∗you found in part (d)? Is E(X) big-ger then, smaller than, or equal to x∗from part (d)? Give an intuitiveexplanation for the relationship between E(X) and x∗in this problem.7. Consider the following discrete random variables:(a) X has a Bernoulli distribution, with P [X = 1] = 2/3, P [X = 0] = 1/3.(an unfair coin toss).(b) X takes the values 1, 2, 3, 4, 5, and 6, with each outcome equally likely,P (X = j) = 1/6 for j = 1, 2, ..., 6. (the result of rolling a fair, six sideddie).(c) X takes the value 1, 2, 3, 4, and 6, with P (X = j) = 1/6 for j = 1, 2, 3, 4,and P (X = 6) = 1/3 (the result of rolling a loaded die that will neverland on 5).For each of the above examples:(a) Graph the probability mass function.(b) Write down the equation for the cdf FX. Graph FX.(c) What is the support of the distribution of X?3(d) What is P (X ≤ 1)?(e) What is P (X ≤ 3)?(f) What is P (X ∈ {1, 3, 5})?(g) What is P (1 < X ≤ 3)?(h) What is P (1 ≤ X ≤ 3)?(i) What is E(X)?8. (Acceptance Sampling.) The following two acceptance rules are being con-sidered for determining whether to take the delivery of a large shipment ofcomponents:Rule 1. A random sample of ten components is checked, and the shipment isaccepted only if none of them is defective.Rule 2. A random sample of twenty components is checked, and the ship-ment is accepted only if not more than one of them is defective.Which of these acceptance rules has the smaller probability of accepting ashipment containing 20% defectives?9. (Sequential Updating) A particular product that you consider for marketingwill either sell well or be a bust. You assess probability 1/3 that it will sellwell and 2/3 that it will be a bust. Consumers will be selected at randomand asked whether they like the product. If the product will sell well, thereis probability .6 that any consumer will say that he or she likes the product.If the product will be a bust, there is probability .06 that any consumer willsay that he or she likes the product.(a) What is the probability