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40 CHAPTER 2. 2 Conditional expectations as random variables Since E X jY = y depends on the actual value of Y , and Y is random, the conditional expectation is a random variable itself. We shall write E fX jY g or E Y X for the random variable de ned by the conditional expectation E fX jY = yg. 25 Suppose Y is a discrete with di erent values on the events A1; A2 ; : : : ; An which form a non-degenerate disjoint partition of the probability space . ; R where mk = Ak X dP=P Ak. In other words, on Ak we have E fX jFg = Ak X dP=P Ak .

1 is usually less accurate. In Chapter 10 we shall see that linear approximation is the best one can get in the all important normal case. Even in non-normal case linear approximations o er quick solutions based on simple second order statistics. In contrast, the non-linear approximations require elaborate numerical schemes to process the empirical data. 15 Lack of memory Conditional probabilities help us to arrive at important classes of densities in modeling. In this section we want to analyze an non-aging device, which characteristics do not change with time.

9 To nd the average of the uniform U 0; 1 distribution, take n1 U1 + : : : + Un. This is the basis of Monte-Carlo methods , Rwhich is a family of related probabilistic methods for computing the integrals. ToR nd ab f x dx we simulate Xj = f a + bUj . P The average n1 nj=1 Xj approximates b,1 a ab f u du for large n. The variance of the n-th approximation is of the order n,1=2 , which is worse than the trapezoidal rule for smooth functions. In return the approximation is insensitive to the smoothness of the integrands, and also to the dimension of the integral.

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Applied probability and stochastic processes by Bryc W.

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