Joint density function

May 2010
27
0
Does anyone know how to calulate the joint density function of X and Z

in this question

Let X and Y be independent standard normal random variable, and
Z (equals) aX (plus) (1-a^2)^1/2 * Y
 

matheagle

MHF Hall of Honor
Feb 2009
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Does anyone know how to calulate the joint density function of X and Z

in this question

Let X and Y be independent standard normal random variable, and
Z (equals) aX (plus) (1-a^2)^1/2 * Y
Z is a normal, since this is linear in X and Y.
Thus just find the mean and variance of Z and you're done.
You can use MGFs, but that silly.
It does prove what I'm saying though.

Oh gave you the marginal of Z.
If you want the joint of X and Z, you need to transform, calc 3 from X,Y to X,Z.
Obtain the jacobian and substitute from one density to the other.