关于连续型随机变量函数概率密度的一个注记

A Note on the Probability Density of Continuous Random Variable Function

文章针对一个连续型随机变量被一类可导函数作用之后形成的新随机变量的概率密度公式计算问题,提出了三个定理进行讨论。通过考虑分段严格单调函数在各分段区间内部的反函数及其定义域,利用概率、反函数、严格单调连续函数以及积分变换法等方法给出了证明。进而完善了文献中连续型随机变量函数的概率密度的各种求解公式。

Aimming at the computing problem of the probability density formula for the new random variable produced after a continuous random variable acted on by a class of derivable functions,three theorems are proposed for discussion.By considering the inverse function and its definition domain of piecewise strict monotone function within each piecewise interval,the proof is given by using the methods of probability,inverse function,strict monotone continuous function,integral transformation method.Furthermore,various solving formulas for the probability density of continuous random variable function in the literature are improved.