The random variables X and Y are independent and Gaussian wi

The random variables X and Y are independent and Gaussian with means of 1 and -3 respectively. They have equal variances of sigma^2 = 1/4. Write each of the probabilities below in terms of Q() functions whose arguments are positive.

Solution

X~N(1,0.5²) Y~N(-3,0.5²) independently

a) P[X>2,Y>-2]=P[X>2]*P[Y>-2] as they are independent

                     =P[(X-1)/0.5>(2-1)/0.5]*P[(Y+3)/0.5>(-2+3)/0.5]

                     =P[Z>2]*P[Z>2] where Z~N(0,1)

                     ={1-P[Z<2]}²={1-Q(2)}²     [answer]

b)P[X>2,Y<-4]=P[X>2]*P[Y<-4]   as they are independent

                     =P[Z>2]*P[Z<(-4+3)/0.5]=P[Z>2]*P[Z<-2]         [P[X>2]=P[Z>2 is from part a)]

                      ={1-P[Z<2]}{1-P[Z<2]}   [as normal distribution is symmetric]

                      ={1-Q(2)}²    [answer]

c)P[0<X<2,Y<-4]=P[0<X<2]*P[Y<-4] as they are independent

                        =P[(0-1)/0.5<(X-1)/0.5<(2-1)/0.5]*P[Z<-2]   [P[Y<-4]=P[Z<-2] from part b)]

                        =P[-2<Z<2]*P[Z<-2]

                        ={P[Z<2]-P[Z<-2]}*P[Z<-2]={P[Z<2]-(1-P[Z<2])}*{1-P[Z<2]} [as normal distribution is symmetric]

                        =[2Q(2)-1]*[1-Q(2)]   [answer]

d) P[0<X<2,-2<Y<-4]=P[0<X<2]*P[-2<Y<-4] as they are independent

                              =[2Q(2)-1]*P[(-2+3)/0.5<Z<(-4+3)/0.5]   [from part c]

                              =[2Q(2)-1]*P[2<Z<-2]=[2Q(2)-1]*[2Q(2)-1] [from part c]

                              =[2Q(2)-1]² [answer]