# Bayesian Probability Theory: Applications in the Physical by von der Linden W., Dose V., von Toussaint U.

By von der Linden W., Dose V., von Toussaint U.

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10 (Mean value of a function of random variables) Suppose there are N random variables X (1) , . . , X(N ) , where the ith random variable is enumerated by ni ∈ Mi . Given a function Y = f X (1) , . . , X (N ) , the mean value of that function is given by ✐ ✐ ✐ ✐ ✐ ✐ “9781107035904ar” — 2014/1/6 — 20:35 — page 20 — #34 ✐ 20 ✐ Basic definitions for frequentist statistics and Bayesian inference Mean value of a function of several random variables f X(1) , . . , X(N ) := ··· n1 ∈M1 nN ∈MN f Xn(1) , .

P (N|nL , L, I) = Nmax Z = N=nmax (N − L)! N! This PMF and its mean and variance can easily be computed numerically. In order to get a better insight into the result, we proceed analytically based on a quite reasonable assumpL and hence N L. Then we can approximate the probability by tion, namely nmax (N − L)! ≈ N −L . N! With the same justification, the normalization constant Z can approximate the following integral: Nmax N −L dN = Z ≈ N=nmax N −L+1 L−1 nmax −→ Nmax Nmax nmax n−L+1 max . L−1 This final result reads −L .

1 Rolling a (fair) die can result in the face values n ∈ {1, 2, 3, 4, 5, 6} with the corresponding probabilities Pn = 16 . The random variable X is equal to the face value. 5. The function Y = f (X) shall define the gain for each outcome. Then the mean f is the average gain. Probability distributions can be characterized in different ways. The most common way is provided by the PMF. Another approach, which is sometimes more convenient, is based on moments. 5 (ith moment of a random variable) The mean of f (X) = Xi is the ith moment of a random variable mi := X i .

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