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. Problem 3 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 an- swers 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. Sub- conscious 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. (i) Determine PK(K), the probability mass function for the number of conscious responses Dave makes in an interval of t minutes. (ii) If we pick any question to which Dave has responded, what is the probability that his answer to that question represents a conscious response? Represents a conscious correct response? (iii) 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? (iv) The papers are to be collected as soon as Dave has completed exactly n responses. Determine the expected value and the probability mass function of the number of questions he will Solution. Write Nc(t) (resp. N(t)) for the number of conscious (resp. unconscious) responses Dave makes in the time-interval [0,t]. We know that Nc and Ns are independent Poisson processes with rate c and s, respectively. (i) By time-homogeneity of Poisson process, K has the same distribution as Nc(t), which is simply Poisson (Act). Therefore (Act)e-Act, if k = 0,1,2, ; PK(k) = 0, k! otherwise. (ii) By merging/splitting of Poisson processes, Similarly, P(the response represents a conscious one) = = + P(the response represents a conscious correct one) = + (iii) By time-homogeneity, we may assume the time-interval is simply [0,t]. Then the event in question be written as {N(t) = r}\{N(t) = s}, and so, may P(r conscious responses and s subconscious responses in a time-interval of length t) = P(N(t) = r, N(t) = s) = P(N(t) = r)P(N(t) = s) = (Act)-het (st) -st. r! s! (iv) Write X for the qaulity of the k-th response Dave makes. (By quality of the response, we mean any of the four possible states in the set {conscious, unconscious} {correct, incorrect}.) By merg- ing/splitting, we know that (k) KEN is an i.i.d. sequence of random variables with Pc correct conscious incorrect 1 - Pc Xk Ps correct unconscious Ac+ As incorrect 1 - Ps Then the probability that the k-th response is correct is given by P(Xk conscious & correct) + P(X = unconscious & correct) = Pcc + Pss + Denoting this probability by q, it follows that the number N of correct responses out of n questions is binomially distributed with parameters n and q. Therefore, with q = Pc+ Pss ' ( E[N]=nq and PN (k) = ()q(1 q)-k, if k {0, . , n}; 0, otherwise.