Question one. a) The number of connections on the internet during two-minute period is given by the following distribution Number of times Proportion i. ii. 0 0.1 1 0.2 2 0.1 3 0.4 4 0.2 Determine the mean number of times a connection is made during a two-minute period ii. Determine the standard deviation of the number of times a connection is made during a two-minute period b) Abdullah is considering an investment in a company. He is interested in buying a number of shares of this company and keeps them for one year. The price of a single share at this time is $16 and the forecasted price a year from today is given in the following table Price of stock in one year Probability i. ii. 15 0.15 16 0.20 17 0.26 18 0.21 19 0.18 Does the information above describe a valid probability distribution function? Justify ii. What is the variance and standard deviation of the stock price a year from today? C) A packing company is inspecting a shipment of grapefruit by cutting 15 grapefruit selected at random. The company will accept all grapefruit if 14 or all of these 15 grapefruit are of satisfactory quality. Suppose that 95% of the grapefruit of the shipment are of satisfactory quality. i. ii. Write the probability distribution of the grapefruits that are of satisfactory quality in the sample of 15 grapefruit. What is the probability that the shipment will be accepted? (I.e. the probability of 14 or 15 good grapefruits out of 15)? d) A bag contains 7 English books and 5 Math books. Suppose that you randomly select 4 books from the bag. Let X is the number of Math books in these 4 books that will be selected from the bag. i. ii. Write the probability distribution of X? What is the probability that you select two Math and two English books? e) Esther must select two Management courses and 3 Math courses. If the catalog contains 5 management and 8 Math courses, how many ways can Esther complete her schedule?