题目
24.设x_(1),x_(2),...,x_(16)是来自N(8,4)的样本，试求下列概率：(1)P(x_((16))>10);(2)P(x_((1))>5).
24.设$x_{1},x_{2},\cdots,x_{16}$是来自N(8,4)的样本，试求下列概率：
(1)$P(x_{(16)}>10)$;
(2)$P(x_{(1)}>5)$.
题目解答
答案
(1) $P(x_{(16)} > 10) = 1 - P(x_{(16)} \leq 10) = 1 - [P(x_1 \leq 10)]^{16}$
由 $x_1 \sim N(8, 4)$，得 $P(x_1 \leq 10) = \Phi(1) \approx 0.8413$，
故 $P(x_{(16)} > 10) \approx 1 - 0.8413^{16} \approx 0.9370$。
(2) $P(x_{(1)} > 5) = [P(x_1 > 5)]^{16}$
由 $x_1 \sim N(8, 4)$，得 $P(x_1 > 5) = 1 - \Phi(-1.5) = \Phi(1.5) \approx 0.9332$，
故 $P(x_{(1)} > 5) \approx 0.9332^{16} \approx 0.3308$。
答案：
(1) $P(x_{(16)} > 10) \approx \boxed{0.9370}$
(2) $P(x_{(1)} > 5) \approx \boxed{0.3308}$