Suppose X N1515 Compute PX 175 Compute PX2 10 Compute P14 X

Suppose X N(15,1.5) Compute P(X 17.5) Compute P(X2 10) Compute P(14 X 18) Compute P( |X-15| 3)

Solution

Normal Distribution
Mean ( u ) =15
Standard Deviation ( sd )=1.5
Normal Distribution = Z= X- u / sd ~ N(0,1)                  

a)
P(X > 17.5) = (17.5-15)/1.5
= 2.5/1.5 = 1.6667
= P ( Z >1.667) From Standard Normal Table
= 0.0478                  
P(X < = 17.5) = (1 - P(X > 17.5)
= 1 - 0.0478 = 0.9522                  

c)
To find P(a < = Z < = b) = F(b) - F(a)
P(X < 14) = (14-15)/1.5
= -1/1.5 = -0.6667
= P ( Z <-0.6667) From Standard Normal Table
= 0.25249
P(X < 18) = (18-15)/1.5
= 3/1.5 = 2
= P ( Z <2) From Standard Normal Table
= 0.97725
P(14 < X < 18) = 0.97725-0.25249 = 0.7248                  

d)
P|X-15|<=3
=> -P(X-15)<=3, P(X-15)<=3
=> P(X-15)>=-3, P(X-15)<=3
=> P>=12, P(X)<=18
=> P(12<=X<=18)

P(X < 12) = (12-15)/1.5
= -3/1.5 = -2
= P ( Z <-2) From Standard Normal Table
= 0.02275
P(X < 18) = (18-15)/1.5
= 3/1.5 = 2
= P ( Z <2) From Standard Normal Table
= 0.97725
P(12 <= X <= 18) = 0.97725-0.02275 = 0.9545