STAT-324Midterm IFebruary 22, 20059:30-10:40AMInstructor: Moo K. ChungAnswer all questions showing all your work neatly.Calculators are allowed. One page note is permittedName:Student ID:Pledge: On my honor, I have neither given nor receivedunauthorized aid on this examination.Signature:Question Points1234Total1. (a) Suppose that A and B are independent events such that the probability that neither o ccursis12and the probability of B is13. Determine the probability of A occurring [5pts].(b) If A and B are independent events, show that Acand B are also independent [5pts].2. A lie detector will show a positive reading (indicates a lie) 10% of the time when a person istelling the truth and 90% of the time when the person is lying. Suppose two p eople are suspectsin a crime and only one is guilty. The guilty one will be lying while the innocent one will betelling the truth.(a) What is the probability that the detector shows positive readings for both suspects [5pts]?(b) What is the probability that a person is lying when the detector shows a negative reading[5pts]?3. A man with n keys wants to op en his door and tries the keys at random. Exactly one key willopen the door.(a) Suppose unsuccessful keys are not eliminated from n keys. Determine the probabilitydistribution of the number of trials until he opens the door [5pts].(b) Suppose unsuccessful keys are eliminated from n keys. Determine the probability distri-bution of the number of trials until he opens the door [5pts].4. We have a probability density f(x) = kx2for 0 < x < 1 and f (x) = 0 otherwise.(a) Compute the variance and the expected value of the distribution [5pts].(b) Determine the 75-th percentile point