The Darth Vader rule

Open access


Using Henstock’s generalized Riemann integral, we show that, for any almost surely non-negative random variable X with probability density function fX and survival function sX(x) := ∫∞ x fX(t) dt, the expected value of X is given by E(Xz) = ∫0 fX(x) dx.

[1] FELLER, W.: An Introduction to Probability Theory and its Applications. Vol. 2. John Wiley & Sons, Inc. New York, 1966.

[2] HENSTOCK, R.: Lectures in the Theory of Integration. World Scientific, Singapore, 1988.

[3] HEWITT, E.: Integration by parts for Stieltjes integrals, Amer.Math.Monthly 67 (1960), 419-423,

[4] KURZWEIL, J.: Generalized ordinary differential equations and continuous dependenceon a parameter, Czechoslovak Math. J. 82 (1957), 418-446.

Tatra Mountains Mathematical Publications

The Journal of Slovak Academy of Sciences

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CiteScore 2017: 0.37

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researchers in the all fields of mathematical research


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