题目
设连续型随机变量 X sim N(3,2^2)，则 P2 < X leq 5 = _____。注：phi(1) = 0.8413，phi(0.5) = 0.6915
设连续型随机变量 $X \sim N(3,2^2)$，则 $P\{2 < X \leq 5\} = \_\_\_\_\_$。
注：$\phi(1) = 0.8413$，$\phi(0.5) = 0.6915$
题目解答
答案
将 $X \sim N(3, 2^2)$ 转化为标准正态变量 $Z = \frac{X - 3}{2}$，则
$P\{2 < X \leq 5\} = P\left\{-0.5 < Z \leq 1\right\} = P\{Z \leq 1\} - P\{Z \leq -0.5\}$
由标准正态分布性质 $P\{Z \leq -0.5\} = 1 - P\{Z \leq 0.5\}$，得
$P\left\{-0.5 < Z \leq 1\right\} = \phi(1) + \phi(0.5) - 1 = 0.8413 + 0.6915 - 1 = 0.5328$
答案： $\boxed{0.5328}$