# 1965 the theory of stochastic processes methuen co ltd

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Unformatted text preview: e is 91/6 49/4 = 70/24. Taking the square root we see that the standard deviation is 1.71. The three possible deviations, in the sense of |X E X |, are 0.5, 1.5, and 2.5 with probability 1/3 each, so 1.71 is indeed a reasonable approximation for the typical deviation from the mean. Example A.19. Geometric distribution. Starting with the sum of the geometric series 1 X (1 ✓) 1 = ✓n n=0 and then di↵erentiating twice and discarding terms that are 0, gives (1 ✓) 2 = 1 X n✓ n 1 and 2(1 ✓) 3 = n=1 Using these with ✓ = 1 EN = EN (N 1) = 1 X n=1 1 X 1 X n(n 2 n=2 p, we see that n(1 p)n n(n 1)(1 1 p = p/p2 = p)n 1 1 p p = 2p 3 (1 p)p = n=2 and hence 1)✓n var (N ) = EN (N = 1) + EN 2(1 p) p +2 p2 p 2(1 p) p2 (EN )2 1 (1 p) = p2 p2 The deﬁnition of expected value generalizes in the usual way to continuous random variables. We replace the probability function by the density function and the sum by an integral Z Eh(X ) = h(x)fX (x) dx (A.19) 217 A.3. EXPECTED VALUE, MOMENTS Example A.20. Uniform distribution...
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