3DECISION MAKINGvariables multiplied by their respective probabilities. If X assumes the values x1, x2, ...., xk withrespective probabilities p1, p2, ...., pk, where ∑i=1kxipi=1, then E(X)=∑i=1kxipi, provided it is finite. For example, let a coin be tossed continuously till the first head appears. The number ofthrows needed to get the first head is a random variable denoted by X, taking the values 1, 2, 3,....ad inf. Further, if the coin is unbiased, the probability of getting a head in a throw is 1/2 andof getting a tail is 1/2. Hence,p[X=k]=(12)k−1∗12,for the throws are made independently of each other and x = k if and only if, a head is obtainedin the kth throw but in none of the earlier throws. The mathematical expression of X is, therefore,E[X]=∑k=1∞k(12)k−1∗12=16[1+2(12)+3(12)2+...]=12(1−12)−2=4.Thus, on an average, 4 throws will be needed to get the first head.c) The expected values of the variables, their respective probabilities and cumulativeprobabilities have been calculated and given in the following table.

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