Question Four (20 Marks) a) A student prepares for an exam by studying a list of 15 problems. She can solve six of them. For the exam, the instructor selects 8 questions at random from the list of fifteen. What is the (6 Marks) probability that the student can solve at least four problems on the exam? b) Suppose that a system contains a certain type of component whose time, in years, to failure is a random variable having an exponential distribution with a mean of 5 years. If 6 of these components are installed in different systems, what is the probability that at least 3 are still functioning at the end of 8 years? c) Let X be a gamma random variable with probability density (7 Marks) function(x)= - exp(-2x), x>0, with a > 2 and 2>0. Prove that the variance of 2 (a) the random variable is X 22 (a-1)(a-2) (7 Marks) Question Five (20 Marks) a) Let X be a discrete random variable with probability mass function(x)= x=1,2,3,. elsewhere Find the probability that X is odd. (4 Marks) b) Suppose that X is a random variable which takes on one of the values 0, 1, 2, 3. If P(X-1)=0.3. P(X-2)=0.2 and P(X=0)=3/(X-3), find the mean and variance of X. (5 Marks) c) The time required to complete an assembly operation is uniformly distributed between 30 and 40 seconds. Find 5) The proportion of assemblies that requires more than 35 seconds to complete The mean and standard deviation of time of assembly (6 Marks) d) The number of automobiles sold weekly at a certain dealership is a random variable with mean 16 and variance 9. Find a lower bound on the probability that next week's sales are between 10 and 22 inclusively. (5 Marks)