Determine the value of c for the distribution of random variable X that has a discrete distribution with the following probability function, f(x) =  cx for x = 1, . . . , 5 0 otherwise [2 marks] Data...

Determine the value of c for the distribution of random variable X that has a discrete distribution with the following probability function, f(x) =  cx for x = 1, . . . , 5 0 otherwise [2 marks] Data Source: Probability and Statistics M. DeGroot & M. Schervish, Addison-Wesley (2002). (b) Let X be the ticket price for an event. supose that the probability distribution of X is: X 100 120 140 160 180 200 p 0.22 0.2 0.18 0.16 0.13 0.11 (i) What is the probability that a randomly selected attendee paid more that $140 for the ticket? [2 marks] (ii) What is the probability that a randomly selected attendee paid less than $160? [2 marks] (iii) Compute the expected value and standard deviation of X. [4 marks] Data Source: Statistics and data with R. Y. Cohen & J. Cohen, Wiley (2008). Question 2 - 6 marks Suppose that a box contains 7 red balls and and 3 blue balls. If 5 balls are selected at random, without replacement, determine the probability function of the number of red balls that will be obtained using (a) Counting methods 3 marks (b) Simulation 3 marks . Data Source: Probability and Statistics M. DeGroot & M. Schervish, Addison-Wesley (2002). 1 Question 3 - 8 marks Suppose that the life expectancy X of each member of a group of people is a random variable having an exponential distribution with parameter ? = 1 50 years. (a) For an individual from this group, compute the probability that: (i) He will survive to 65, 2 marks (ii) He will live to be at least 70 years old, given that he just celebrated his 40th birthday, 3 marks (b) For what value of c is P(X > c) = 1 2 ? 3 marks f(x|?) = ?e-?x ? > 0, x > 0 Data Source: A Course in Mathematical Statistics. G Roussas, Academic Press (1997). Question 4 - 6 marks A uniform distribution is defined with density function f(x; a, ß) = 1 (ß - a) ß > a and x ? (a, ß) Show that (a) E(X) = a + ß 2 [2 marks] (b) var(X) = (ß - a) 2 12 [4 marks] Data Source: A Course in Mathematical Statistics. G Roussas, Academic Press (1997). Question 5 - 6 marks Let X be a random variable with probability density function f(x) = ?e-?(x-a) x > a (a) Find its Moment Generating Function MX(t) for those t’s that exist. [3 marks] (b) Calculate E(X) [1 mark] (c) Calculate s 2 (X) [2 marks] Data Source: A Course in Mathematical Statistics. G Roussas, Academic Press (1997)