随机变量的函数
设\(X\)为一一维随机变量，\(f: \R \to \R\)为一函数，那么\(f(X)\)的期望以及分布情况会是什么样的呢？
我们这里只讨论\(f\)是单调函数的情况，令\(Y = f(X)\)，那么 \[ P_Y(y) = P_Y(Y \le y) = P_X(f(X) \le y) \]
若\(f\)单调递增， \[ \begin{gather} P_Y(f(X) \le y) = P_X(X \le f^{-1}(y)) = P_X(f^{-1}(y)) \\ \nonumber \\ \begin{split} p_Y(y) &= \frac{\partial P_Y(Y \le y)}{\partial y} \\ &= \frac{\partial P_X(f^{-1}(y))}{\partial y} \\ &= p_X(f^{-1}(y)) \cdot (f^{-1})^\prime (y) \\ &= p_X(f^{-1}(y)) \cdot |(f^{-1})^\prime (y)| \\ \end{split} \end{gather} \]
若\(f\)单调递减， \[ \begin{gather} P_Y(f(X) \le y) = P_X(X \ge f^{-1}(y)) = 1 - P_X(X \le f^{-1}(y)) = 1 - P_X(f^{-1}(y)) \\ \nonumber \\ \begin{split} p_Y(y) &= \frac{\partial P_Y(Y \le y)}{\partial y} \\ &= \frac{\partial [1 - P_X(f^{-1}(y))]}{\partial y} \\ &= -p_X(f^{-1}(y)) \cdot (f^{-1})^\prime (y) \\ &= p_X(f^{-1}(y)) \cdot |(f^{-1})^\prime (y)| \\ \end{split} \end{gather} \]
总而言之，\(p_Y(y) = p_X(f^{-1}(y)) \cdot |(f^{-1})^\prime (y)|\)。