Introduction a La Mecanique Quantique - 1Er Cycle by Jean Hladik Et Michel Chrysos

By Jean Hladik Et Michel Chrysos

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Consider the sum S = Xl + ... + X N • Then s(t) = Et S = E[E(tSI N)] = E~(t) = N(X{t)). Similarly ES = E[E(SIN)] = E(NEX) = ENEX, 36 II. Multivariate Random Variables and by (11) + Var E(SIN) E(N Var X) + Var(NEX) = EN Var X + (EX)2 Var N. Var S = E Var(SIN) = This example will be useful in our discussion of branching chains. Orthogonal Projections We describe here an alternative approach to conditioning, useful in the mean square setting. Theorem II. Let C be a closed convex set in a Hilbert space Yt'.

N n~1 ~I :2 I-I

This is elaborated later, when we interpret E(YIX) as an orthogonal projection. Conditioning is most useful when one is studying compound random variables for which some parameters are themselves random variables. For example, let Xl' X 2 , ••• be independent identically distributed nonnegative integer-valued random variables with common generating function x. Let N be a nonnegative integer-valued random variable, independent of the Xns, with generating function N. Consider the sum S = Xl + ...

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