设随机变量$X$的概率密度
$$
f(x)=\left\{\begin{array}{c}
x, 0 \leq x \leq 1 \\
c-x, 1 < x \leq 2, \quad \text { 记 } Y=2 X-1 , 0 \text {, 他他 } \\
0, \text { 其 }
\end{array}\right.
$$
记$Y=2X-1$ 求
(1) 常数 $c$;
(2) $P\left(X < \frac{7}{6}\right)$;
(3) $Y$ 的密度函数 $f_Y(y)$
【答案】 (1) $1=\int_0^1 x d x+\int_1^2(c-x) d x, \quad c=2$
(2) $P\left(X < \frac{7}{6}\right)=\int_0^1 x d x+\int_1^{7 / 6}(2-x) d x=\frac{47}{72}$
(3)
$$
\begin{gathered}
F_Y(y)=P(Y \leq y)=P\left(X \leq \frac{y+1}{2}\right)=F_X\left(\frac{y+1}{2}\right) \\
f_Y(y)=\frac{1}{2} f_X\left(\frac{y+1}{2}\right)=\left\{\begin{array}{c}
(y+1) / 4,-1 \leq y \leq 1 \\
(3-y) / 4, \quad 1 < y \leq 3 \\
0, \text { 其他 }
\end{array}\right.
\end{gathered}
$$


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