A taxi travels between four villages, , situated at the corners of a rectangle. The four roads connecting the villages follow the sides of the rectangle; the distance from to and to is 5 miles and from to and to miles. After delivering a customer the taxi waits until the next call then goes to pick up the new customer and takes him to his destination. The calls may come from any of the villages with probability and each customer goes to any other village with probability . Naturally, when travelling between a pair of adjacent corners of the rectangle, the taxi takes the straight route, otherwise (when it travels from to or to or vice versa) it does not matter. Distances within a given village are negligible. Let be the distance travelled to pick up and deliver a single customer. Find the probabilitites that takes each of its possible values. Find the expected value and the variance Var .
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