20. Runs. Consider a sequence of N dependent trials, and let X; be 1 or 0 as the ith trial is a success or failure. Suppose that the sequence has the Markov property! P{X; = 1IxI , .. ·, xi- d = P{X; = 1lxi-d and the property of stationarity according to which P{X; = I} and P{X; = 1Ixi_l} are independent of i, The distribution of the X's is then specified by the probabilities PI = P{X; = 11X;-1 = I} and Po = P{X; = 11X;-1 = O} and by the initial probabilities 'lT1 = P{XI = I} and 'ITo = 1 - 'lT1 = P{XI = O}. (i) Stationarity implies that Po -- , 'lT1 = Po + ql ql 'ITo = Po + ql . (ii) A set of successive outcomes Xi,Xi+I" "'Xi+} is said to form a run of zeros if X i = Xi+1 = ... = xi+} = 0, and Xi_I = lor i = 1, and Xi+}+1 = 1 or i + j = N. A run of ones is defined analogously. The probability of any particular sequence of outcomes (XI' ... , XN) is 1 Po + ql pgpi-Vqjq(j'-u,

where m and n denote the numbers of zeros and ones, and u and v the numbers of runs of zeros and ones in the sequence.