题目
已知X～N(1,4),计算:P(X<1) = _ ;P(X<2) = _ ;P(2<X<3) = _ ;P(X>0) = _ ;P( ｜X｜ >2.5) = _.
已知X～N(1,4),计算:
P(X<1) = _ ;
P(X<2) = _ ;
P(2<X<3) = _ ;
P(X>0) = _ ;
P( ｜X｜ >2.5) = _.
题目解答
答案
已知X～N(1,4)，则μ=1，σ=2。
(1) P(X<1) = P(Z<(1-1)/2) = P(Z<0) = 0.5
(2) P(X<2) = P(Z<(2-1)/2) = P(Z<0.5) = 0.6915
(3) P(2<X<3) = P((2-1)/2<Z<(3-1)/2) = P(0.5<Z<1) = P(Z<1)-P(Z<0.5) = 0.8413-0.6915 = 0.1498
(4) P(X>0) = 1-P(X<0) = 1-P(Z<(-1)/2) = 1-P(Z<-0.5) = 1-0.3085 = 0.6915
(5) P(|X|>2.5) = P(X<-2.5)+P(X>2.5) = P(Z<(-2.5-1)/2)+P(Z>(2.5-1)/2)
= P(Z<-1.75)+P(Z>0.75)
= 0.0401+0.2266
= 0.2667
解析
步骤 1：计算P(X<1)
P(X<1) = P(Z<(1-1)/2) = P(Z<0) = 0.5
步骤 2：计算P(X<2)
P(X<2) = P(Z<(2-1)/2) = P(Z<0.5) = 0.6915
步骤 3：计算P(2P(2步骤 4：计算P(X>0)
P(X>0) = 1-P(X<0) = 1-P(Z<(-1)/2) = 1-P(Z<-0.5) = 1-0.3085 = 0.6915
步骤 5：计算P(|X|>2.5)
P(|X|>2.5) = P(X<-2.5)+P(X>2.5) = P(Z<(-2.5-1)/2)+P(Z>(2.5-1)/2) = P(Z<-1.75)+P(Z>0.75) = 0.0401+0.2266 = 0.2667
P(X<1) = P(Z<(1-1)/2) = P(Z<0) = 0.5
步骤 2：计算P(X<2)
P(X<2) = P(Z<(2-1)/2) = P(Z<0.5) = 0.6915
步骤 3：计算P(2
P(X>0) = 1-P(X<0) = 1-P(Z<(-1)/2) = 1-P(Z<-0.5) = 1-0.3085 = 0.6915
步骤 5：计算P(|X|>2.5)
P(|X|>2.5) = P(X<-2.5)+P(X>2.5) = P(Z<(-2.5-1)/2)+P(Z>(2.5-1)/2) = P(Z<-1.75)+P(Z>0.75) = 0.0401+0.2266 = 0.2667