Dave is taking a multiple-choice exam. You may assume that the number of questions is infinite. Simultaneously, but independently, his conscious and subconscious faculties are generating answers for him, each in a Poisson manner. (His conscious and subconscious are always working on different questions.) Conscious responses are generated at a rate of c responses per minute. Subconscious responses are generated at a rate of s responses per minute. 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. (a) Determine pK (k), the probability mass function for the number of conscious responses Dave makes in an interval of t minutes. (b) If we pick any question to which Dave has responded, what is the probability that his answer to that question: 7 (i) Represents a conscious response. (ii) Represents a conscious correct response. (c) 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? (d) Determine the transform for the probability density function for random 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. (e) Determine the probability mass function for the total number of responses up to and including his third conscious response. (f) The papers are to be collected as soon as Dave has completed exactly n responses. Determine: (i) The expected number of questions he will answer correctly (ii) The probability mass function for L, the number of questions he answers correctly. (g) Repeat part (f) for the case in which the exam papers are to be collected at the end of a fixed interval of t minutes.