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Q8 Consider the following circuit of switches P D p A P1 B . The circuit has three branches (a top(t), a middle(m) and a bottom (b) branch). . A current can flow through a switch if it is closed. If it is open, then no current can flow through it. . The probabilities that switches are closed are given in the drawing. (That is, the switch in the middle branch is closed with probabulity p1, and the switches in the other two branches are closed with probability p . For example, since the probability that the switch in the middle branch is closed is p1, the probability that a current flows from A to B along the middle branch is p1. . Similarly, the probability that all three switches are closed in the top branch is p3, and so a current will flow from A to B along the top branch with probability p3 (and a current will not flow along this branch with probability 1 - p3). . Notice that p1 / P. Define q = 1 - p and q1 = 1 - P1. Assume that the switches are independent, and that Xm is the number of closed switches along the middle branch, X, is the number of closed switches along the top branch, and X, is the number of closed switches along the bottom branch. Answer the following questions: (a ) What is the expectation of Xm? [2] (b) What is the distribution of X+? [2] (c) What is the expectation of X? [2] (d) What is the distribution of Xb? [1] (e) What is the expectation of X ? [1] (f ) What is the expectation of the number of open switches? [2] (g) What is the variance of the number of open switches? [2] (h) What is the probability that a current flows between A and B? [5] [17]