Abstract
Finding expected values of distributions is one of the main tasks of any probabilistic analysis. The expected value in the narrower sense of the average (mean), which is a measure of distribution location, is introduced first, followed by the related concepts of the median and distribution quantiles. Expected values of functions of random variables are presented, as well as the variance as the primary measure of the distribution scale. The discussion is extended to moments of distributions (skewness, kurtosis), as well as to two- and d-dimensional generalizations. Finally, propagation of errors is analyzed.
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- 1.
A function is defined to be convex if the line segment between any two points on the graph of the function lies above the graph.
- 2.
A random vector \({{\varvec{X}}}=(X_1,X_2,\ldots ,X_m)\) with a distribution function \(F_{{\varvec{X}}}(x_1,x_2,\ldots ,x_m)\) and a random vector \({{\varvec{Y}}}=(Y_1,Y_2,\ldots ,Y_n)\) with a distribution function \(F_{{\varvec{Y}}}(y_1,y_2,\ldots ,y_n)\) are mutually independent if \(F_{{{\varvec{X}}},{{\varvec{Y}}}}(x_1,x_2,\ldots ,x_m,y_1,y_2,\ldots ,y_n) = F_{{\varvec{X}}}(x_1,x_2,\ldots ,x_m)F_{{\varvec{Y}}}(y_1,y_2,\ldots ,y_n)\). This is an obvious generalization of (2.20) and (2.24).
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Širca, S. (2016). Expected Values. In: Probability for Physicists. Graduate Texts in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-31611-6_4
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DOI: https://doi.org/10.1007/978-3-319-31611-6_4
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