Question

Suppose that X and Y are standard bivariate normal with correlation ρ = 0.4. You will need to use the above table and compute each of the following:

(a) P(X ≥ 1).

(b) P(X ≥ 1 | Y = 1).

(c) P(X ≥ 1 | Y = −1).

x	0.6547	1	1.5275
Φ(x)	0.7437	0.8413	0.9367

Answer 1

X and Y  are standard bivariate normal with correlation ρ = 0.4.

Therefore X follows Standard Normal distribution.

Therefore P( X >= 1) = 1 - P(X <= 1) = 1 - 0.8413 =

b) The mean and standard deviation of conditional distribution X on Y is as follows :

Where is the mean of standard normal variate of x= 0

is the mean of standard normal variate of y = 0

is the mean of standard normal variate of x= 1

is the mean of standard normal variate of x= 1

and = 0.4

Plug this values in the formula of conditional mean we get

Therefore for y = 1 ,

and variance =

therefore standard deviation =

P(X >=1|Y = 1) = 1 - P(X < 1|Y = 1) ........( 1 )

Lets find Z score

Therefore P(X < 1|Y = 1) = 0.7437

Plug this value in equation ( 1)

P(X >=1|Y = 1) = 1 - 0.7437 = 0.2563

(c) P(X ≥ 1 | Y = −1) = 1 - P(X <1 | Y = −1) -----------( 2 )

Mean = 0.4* y = 0.4 * ( - 1 ) = - 0.4

standard deviation is same as part b) = 0.9165

therefore Z score for x = 1 is

Therefore ,  P(X <1 | Y = −1) = 0.9367

Put this value in equation ( 2 ) , so we get:

P(X ≥ 1 | Y = −1) = 1 - 0.9367 = 0.0633