Random variables

Suppose that (X, Y) has a bivariate Gaussian distribution with means ux and uy, variances or and ov, and correlation coefficient p. Show that Y can be written Y poy ( X - MX) + MY + Z (1) OX where Z is Gaussian with mean zero and variance (1 - p2 )? and is inde- pendent of X. Hint: This problem is much easier than it first appears. Simply define Z via (1), argue that it has a Gaussian distribution, and then compute its mean, variance, and covariance with X using the expectation operator.Let X1, ...,Xn be a random sample from the inverse Gaussian distribution with pdf A 1/2 -Me-)2 f(z| 4, A ) = e 0