X1 and X2 are independent random variabes with X1N42 X2N21 a

X1 and X2 are independent random variabes, with X1~N(4,2), X2~N(2,1).

a) find P(X2 >= X1)

b) P(X1-4<X2<X1-1)

Solution

a)  P(X2 >= X1) = p(X2 >=4) =

For x = 4, z = (4 - 2) / 1 = 2

Hence P(x >= 4) = P(z >= 2) = [area to the left of 2]

= 0.9772

B)  P(X1-4<X2<X1-1) = P(0<X<3) =

For x = 1 , z = (0 - 2) / 1 =-2 and for x = 3, z = (3 - 2) / 1 = 1

Hence P(0< x < 3) = P(-2 < z < 1) = [area to the left of z = 1] - [area to the left of -2]

= 0.8413 - 0.0228 = 0.8185