大二下：概率论与数理统计复习 2.随机变量及其分布之习题加练

大二下：概率论与数理统计复习 导航页：https://blog.csdn.net/COCO56/article/details/100152856

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4. 已知随机变量X的概率密度求$f_Y(y)$

$已知随机变量X的概率密度函数f_X(x)=\{\begin{array}{rlr}& \frac{1}{5}{e}^{-\frac{x}{5}}& x>0\\ & 0,& x\le 0\end{array}$$,则Y=2X+1的密度函数f_Y(y)=\underline{\{\begin{array}{rlr}& \frac{1}{10}{e}^{-\frac{y-1}{10}},& y>1\\ & 0,& y\le 1\end{array}}.$

$解：$
$\because F_Y(y)=P(Y\le y)=P(2X+1\le y)=P(X\le \frac{y-1}{2})=F_X(\frac{y-1}{2})$
$\therefore f_Y(y)=f_X(\frac{y-1}{2})\times (\frac{y-1}{2}{)}^{\prime }=\frac{1}{2}{f}_X(\frac{y-1}{2})$
$=\{\begin{array}{rlr}& \frac{1}{2}\times \frac{1}{5}{e}^{-\frac{\frac{y-1}{2}}{5}},& \frac{y-1}{2}>0\\ & 0,& \frac{y-1}{2}\le 0\end{array}$
$=\{\begin{array}{rlr}& \frac{1}{10}{e}^{-\frac{y-1}{10}},& y>1\\ & 0,& y\le 1\end{array}$