题目
设随机变量backsim N(2,9), circled (1)(0.5)=0.6915 circled (p)(1)=0.8413，则backsim N(2,9), circled (1)(0.5)=0.6915 circled (p)(1)=0.8413_________;backsim N(2,9), circled (1)(0.5)=0.6915 circled (p)(1)=0.8413_________.
设随机变量
，则
_________;
_________.
题目解答
答案
∵随机变量
∴
∴
∴
∵
∴
故正确答案填的是
。
解析
步骤 1：确定正态分布的参数
随机变量$X$服从正态分布$N(2,9)$，其中均值$\mu=2$，方差$\sigma^2=9$，因此标准差$\sigma=3$。
步骤 2：计算$P(2\leqslant X\leqslant \dfrac {7}{2})$
$P(2\leqslant X\leqslant \dfrac {7}{2})=P(X\leqslant \dfrac {7}{2})-P(X\leqslant 2)$
$P(X\leqslant \dfrac {7}{2})=\Phi(\dfrac {\dfrac {7}{2}-2}{3})=\Phi(\dfrac {1}{2})=0.6915$
$P(X\leqslant 2)=\Phi(\dfrac {2-2}{3})=\Phi(0)=0.5$
$P(2\leqslant X\leqslant \dfrac {7}{2})=0.6915-0.5=0.1915$
步骤 3：计算$P(X\geqslant 2)$
$P(X\geqslant 2)=1-P(X\lt 2)=1-\Phi(\dfrac {2-2}{3})=1-\Phi(0)=1-0.5=0.5$
随机变量$X$服从正态分布$N(2,9)$，其中均值$\mu=2$，方差$\sigma^2=9$，因此标准差$\sigma=3$。
步骤 2：计算$P(2\leqslant X\leqslant \dfrac {7}{2})$
$P(2\leqslant X\leqslant \dfrac {7}{2})=P(X\leqslant \dfrac {7}{2})-P(X\leqslant 2)$
$P(X\leqslant \dfrac {7}{2})=\Phi(\dfrac {\dfrac {7}{2}-2}{3})=\Phi(\dfrac {1}{2})=0.6915$
$P(X\leqslant 2)=\Phi(\dfrac {2-2}{3})=\Phi(0)=0.5$
$P(2\leqslant X\leqslant \dfrac {7}{2})=0.6915-0.5=0.1915$
步骤 3：计算$P(X\geqslant 2)$
$P(X\geqslant 2)=1-P(X\lt 2)=1-\Phi(\dfrac {2-2}{3})=1-\Phi(0)=1-0.5=0.5$