Another formula for the expectation of a function of a random variable and its application
In this paper,we discuss the issue of mathematical expectation of functions of random variables,espe-cially when the random variables are non-negative and the functions are strictly monotonically increasing positive first-order differentiable functions on the non-negative real number interval.The article proposes a new formula for calculating the mathematical expectation of such functions,whose integral expression is related to the tail probability of the non-negative random variable and the derivative of the function.The article first presents two formulas for calculating expectations by utilizing Fubini's theorem and variable substitution methods.Then,two application examples are provided.These applications demonstrate how to apply the theorem to specific func-tions.
non-negative random variablea function of a random variableexpectation
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