【ID】630 【题型】解答题 【类型】考研真题 【来源】1993年全国硕士研究生入学统一考试数学一试题

$$F_{X}(x)=\left\{\begin{array}{ll} \frac{1}{2}, & 0 < x < 2 \\ 0, & \text { 其它 } \end{array} .\right.$$

\begin{aligned} F_{Y}(y) &=P\{Y \leq y\}=P\left\{X^{2} \leq y\right\}=P\{-\sqrt{y} \leq X \leq \sqrt{y}\} \\ &=\int_{-\sqrt{y}}^{\sqrt{y}} F_{X}(x) d x=\int_{-\sqrt{y}}^{0} 0 d x+\int_{0}^{\sqrt{y}} \frac{1}{2} d x=\frac{\sqrt{y}}{2} \end{aligned}

$$F_{Y}(y)= \begin{cases}0, & y \leq 0 \\ \frac{\sqrt{y}}{2}, & 0 < y < 4 \\ 1, & y \geq 4\end{cases}$$

$$f_{Y}(y)=F_{Y}^{\prime}(y)=\left\{\begin{array}{ll} \frac{1}{4 \sqrt{y}}, & 0 < y < 4 \\ 0, & \text { 其他 } \end{array} .\right.$$

$h^{\prime}(y)=\frac{1}{2 \sqrt{y}}$ 恒不为零, 因此, 由连续型随机变量函数的密度公式, 得到随机变量 $Y$ 的概率

$$f_{Y}(y)=\left\{\begin{array}{cc} \left|h^{\prime}(y)\right| f_{X}[h(y)], 0 < y < 4 \\ 0, & \text { 其他 } \end{array}=\left\{\begin{array}{cc} \frac{1}{2 \sqrt{y}} \cdot \frac{1}{2}, 0 < y < 4, \\ 0, & \text { 其他, } \end{array}=\left\{\begin{array}{cc} \frac{1}{4 \sqrt{y}}, & 0 < y < 4, \\ 0, & \text { 其他. } \end{array}\right.\right.\right.$$