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By Abdulle A., Wanner G.

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It is unique except for a set A^ of images of Z which occurs with zero probability P{ZeN] = 0. Here /z is a real-valued function, defined on the real line h: [R^ fR, and we write E{X\Z = z] = h{z). 75) This gives a precise meaning to the expectation of X under the condition that the random variable Z takes on the value z. It is a typical example of a condition that occurs for continuous random variables with probability zero in general. Applying these results to the conditional probability P ( X 6 5 |Z), the conditional distribution can be expressed via an almost surely unique function Q Px{B\ ^){co) = Qx(B\Ziaj)).

No global statement referring to the largest value of the probability density on the state space is implied. Let us consider a few instructive examples. The Gaussian Distribution. A random variable is said be Gaussian or normally distributed if its probabihty density is given by p{x) = [i2n)'''or'exp(- ^"""J^'V (2-38) often denoted by: X is N(m, cr^). 39) E{(SXf}=a^. 38) that a Gaussian variable is completely characterized by its first two moments. 1 Probability Triple and Random Variables E{{dXy]={^ ^ ^ \{r-\)\\G' '^^ r^l ^^^,.

This is expressed by the fact that the a field generated by all sets of the form X ~ ^ ( ( - oo, x]), x e fR, and denoted by j / {X) is in general only a subfield of j / , ^ {X) C j / . A simple random variable in the case of the gas container mentioned above is the speed 11; | of a gas molecule at a given instant of time, if j / = ^ {Q). , the number of elementary outcomes, is finite, the G field of events can be chosen to be the power set and is indeed the natural choice as remarked above. 8) is trivially satisfied in this case.

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