a) Suppose that X is a random variable for which E(X)=2, E(X)=8 and E(X)=12. Find the value of the third central moment of X (4 Marks) b) In a certain manufacturing process it is known that, on average, 2 in every 100 items are defective. What is the probability that the sixth item inspected is the first defective item found? (5 Marks) c) The number of messages sent to a computer bulletin board is a Poisson random variable with a mean of 8 messages per hour. i) What is the probability that 6 messages are received in one hour? What is the probability that 16 messages are received in one and a half hours? ii) iii) What is the probability that at least four messages are received in one-half hour? (9 Marks) Question Three (20 Marks) The sick-leave time of employees in a firm in a month is normally distributed with a mean of 100 hours and a standard deviation of 20 hours. Find the probability that the sick-leave time for next month will be i) Less than 121 hours ii) More than 89 hours. iii) Between 63 and 133 hours? (4 Marks) (5 Marks) (5 Marks) b) The managers of an electronics firm estimate that 70% of the new products they market will be successful. If the company markets 120 products over the next five years, what is the probability that more than 86 but less that 100 new products will be successful?(6 Marks)