the xycoordinates of points in a plane are characterized by

the xy-coordinates of points in a plane are characterized by independent x and y values which are both Gaussian with zero mean and variance 4.

a) find the probability that a point (x,y) lies between a radius of 0.5 and 1.0 about the origin

b) consider only the x coordinate, what is P(0<x<0.4|0<x<0.6)

Solution

a) mean=0 and s.d=2.....
P ( 0.5<x<1 and 0.5 <y<1) = p( ( 0.5<x<1) * p(0.5 <y<1) [as x and y are independent ]
= p ( 0.5/2 < z < 1/2) * p ( 0.5/2 <z < 1/2 ) = [ p(z<0.5) - p(z<0.25) ] * [  p(z<0.5) - p(z<0.25) ]........

= [ 0.6914625 - 0.5987063 ] * [ 0.6914625 - 0.5987063 ] = 0.008603702

b) p( 0<x <0.4 | 0<x<0.6) = p (  0<x <0.4 and 0<x<0.6) / p ( 0<x<0.6 )...=p( 0 < x<0.4 ) / p ( 0<x<0.6 )...
= p ( 0 < z < 0.4/2 ) / p ( 0 < z < 0.6/2 ) = (0.5792597-0.5) / (0.6179114-0.5) = 0.0792597 / 0.1179114 = 0.6721971.........