求解涉及随机变量的问题
本节将介绍如何求解涉及随机变量的问题，特别是随机变量函数的反推问题。当已知函数随机变量的期望值和方差时，如何推导出原随机变量的期望值和方差。
This section introduces how to solve problems involving random variables, particularly inverse problems with functions of random variables. When the expected value and variance of a function random variable are known, how to deduce the expected value and variance of the original random variable.
通过学习本节内容，您将掌握线性变换的反推技巧，理解函数随机变量与原随机变量统计量的关系。这些技能对于解决复杂的概率统计问题至关重要。
By studying this section, you will master the techniques for inverse problems with linear transformations and understand the relationship between the statistics of function random variables and the original random variables. These skills are crucial for solving complex probability and statistics problems.
如果X是随机变量，g是函数，那么Y = g(X)也是随机变量。可以利用Y的统计量推导出X的统计量。
If X is a random variable and g is a function, then Y = g(X) is also a random variable. The statistics of Y can be used to deduce the statistics of X.
对于线性函数Y = aX + b，已知Y的期望值和方差可以直接推导出X的期望值和方差。
For linear function Y = aX + b, knowing the expected value and variance of Y allows direct deduction of the expected value and variance of X.
函数g必须是一一对应的，反推才有意义。线性函数通常满足这个条件。
Function g must be one-to-one for inversion to make sense. Linear functions usually satisfy this condition.
\[\mathrm{E}(X) = \frac{\mathrm{E}(Y) - b}{a}\]
对于 \(Y = aX + b\)
\[\operatorname{Var}(X) = \frac{\operatorname{Var}(Y)}{a^2}\]
对于 \(Y = aX + b\)
\[\sigma_X = \frac{\sigma_Y}{|a|}\]
对于 \(Y = aX + b\)