1.4 Dave is taking a multiple-choice exam. You may assume that the number of questions is infinite. Simultaneously, but independently, his conscious and subconscious facilities are generating answers for him, each in a Poisson manner. (His conscious and subconscious are always working on different questions.) Average rate at which conscious responses are generated = \ responses/min Average rate at which subconscious responses are generated = \, responses/min Each conscious response is an independent Bernoulli trial with probability pc of being correct. Similarly, each subconscious response is an independent Bernoulli trial with probability ps of being correct. Dave responds only once to each question, and you can assume that his time for recording these conscious and subconscious responses is negligible. 1. Determine p(k), the probability mass function for the number of conscious responses Dave makes in an interval of T minutes. 2. If we pick any question to which Dave has responded, what is the probability that his answer to that question: (a) Represents a conscious response (b) Represents a subconscious response 3. If we pick an interval of T minutes, what is the probability that in that interval Dave will make exactly r conscious responses and exactly s subconscious responses. 4. Determine the moment generating function for the probability density function for ran- dom variable X, where X is the time from the start of the exam until Dave makes his first conscious response which is preceded by at least one subconscious response. 5. Determine the probability mass function for the total number of responses up to and including his third conscious response. 6. The papers are to be collected as soon as Dave has completed exactly N responses. Determine: (a) The expected number of questions he will answer correctly (b) The probability mass function for L, the number of questions he answers correctly 7. Repeat part (f) for the case in which the exam papers are to be collected at the end of a fixed interval T minutes.