题目
19. (17.0分) 已知随机变量X的概率密度为 f(x)={}kx+1, & 0le xle 20, & 其他.，求(1)参数k的值; (2)分布函数F(x);(3)P(-(1)/(2)<(1)/(2));(4)E(X),D(X).
19. (17.0分) 已知随机变量X的概率密度为
$f(x)=\left\{\begin{matrix}kx+1, & 0\le x\le 2\\0, & 其他\end{matrix}\right.，求(1)参数k的值;$
(2)分布函数F(x);(3)P(-$\frac{1}{2}$<$\frac{1}{2}$);(4)E(X),D(X).
题目解答
答案
(1) **求参数 $k$**
由概率密度函数在全定义域积分等于1：
\[
\int_0^2 (kx + 1) \, dx = 1 \implies k = -\frac{1}{2}
\]
(2) **求分布函数 $F(x)$**
\[
F(x) = \begin{cases}
0, & x < 0 \\
-\frac{1}{4}x^2 + x, & 0 \le x \le 2 \\
1, & x > 2
\end{cases}
\]
(3) **求 $P\left( -\frac{1}{2} < X < \frac{1}{2} \right)$**
\[
P\left( -\frac{1}{2} < X < \frac{1}{2} \right) = F\left( \frac{1}{2} \right) - F(0) = \frac{7}{16}
\]
(4) **求 $E(X)$ 和 $D(X)$**
\[
E(X) = \frac{2}{3}, \quad D(X) = \frac{2}{9}
\]
\[
\boxed{
\begin{array}{ll}
(1) & k = -\frac{1}{2} \\
(2) & F(x) = \begin{cases}
0, & x < 0 \\
-\frac{1}{4}x^2 + x, & 0 \le x \le 2 \\
1, & x > 2
\end{cases} \\
(3) & \frac{7}{16} \\
(4) & E(X) = \frac{2}{3}, \quad D(X) = \frac{2}{9}
\end{array}
}
\]