(a)What is the PMF of the number of the modems in use at a given time?(b) Repeat part (a) by approximating the PMF of the number of customers that need a connection with a Poisson PMF.13. A total of 46 percent of voters in a certain city classify themselves as Independents where as 30 percent classify themselves as Liberals and 24 percent as Conservatives. In a recent local election 35 percent of the Independents 62 percent of the Liberals and 58 percent of the Conservatives voted. A voter is chosen at random. Given that this person voted in the local election what is the probability that he or she is(a) an Independent?(b) a Liberal?(c) a Conservative?(d) What fraction of voters participated in the local election?19. Two balls are chosen randomly from an urn containing 8 white 4 black and 2 orange balls. Suppose that we win $2 for each black ball selected and we lose $1 for each white ball selected. Let X denote our winnings. What are the possible values of X and what are the probabilities associated with each value? Plot the Probability Mass Function (PMF) of X.20. Two fair dice are rolled. Let X equal the product of the 2 dice. Compute P(X = i) for i = 1; 2.23. A total of 4 buses carrying 148 students from the same school arrives at a football stadium. The buses carryrespectively 40 33 25 and 50 students. One of the students is randomly selected. Let X denote the number of students that were on the bus carrying this randomly selected student. One of the 4 bus drivers is also randomly selected. Let Y denote the number of students on her bus.(a) Which of E[X] or E[Y ] do you think is larger? Why?(b) Compute E[X] and E[Y ].26. A box contains 5 red and 5 blue marbles. Two marbles are withdrawn randomly. If they are the same colorthen you win $1.10; if they are di_erent colors then you lose $1.00. Calculate(a) the expected value of the amount you win;(b) the variance of the amount you win.27. If E[X] = 1 and Var(X) = 5 _find:(a) E[(2 + X)exponent of 2];(b) Var(4 + 3X).