A Probabilistic Analogue of The Mean Value Theorem
Let and be non-negative random variables such that and (i.e. is smaller than in the usual stochastic order). Then there exists an absolutely continuous non-negative random variable having probability density function
Let be a measurable and differentiable function such that and are finite, and let its derivative be measurable and Riemann-integrable on the interval for all . Then, is finite and
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