Let Z1 and Z2 be independent n(0,1) random variables, and define new random variables X and Y by

where aX, bX, cX, aY, bY, and cy are constants.
(a) Show that

(b) If we define the constants aX, bX, cX, aY, bY, and cY by

where μX, μY, σ2X, σ2Y, and p are constants, -1

(c) Show that (X, Y) has the bivariate normal pdf with parameters μX, μY, σ2X, σ2Y and p.
(d) If we start with bivariate normal parameters μX, μY, σ2X, σ2Y and p, we can define constants aX, bX, cX, aY, bY, and cY as the solutions to the equations

Show that the solution given in part (b) is not unique by exhibiting another solution to these equations. How many solutions are there?

Transcribed Image Text:

2 + cy, Cov(X, Y) = axay + bx by. 블eov, 1+ρ br=-