Find the joint PDF of X, Y, Z.

Question

Let X, Y, Z be r.v.s such that X ⇠ N (0, 1) and conditional on X = x, Y and Z are i.i.d. N (x, 1).

(a) Find the joint PDF of X, Y, Z.

(b) Find the joint PDF of Y and Z. You can leave your answer as an integral, though the integral can be done with some algebra (such as completing the square) and facts about the Normal distribution.

Is my reasoning correct?

I did the following:

a) $f(X=x,Y=y,Z=z)=f(X=x)f(Y=y|X=x)f(Z=z|X=x,Y=y)=f(X=x)f(Y=y|X=x)f(Z=z|X=x)=$

$$\frac{1}{2\sqrt \pi} e^{-x^2/2} \, \frac{1}{2\sqrt \pi} e^{-(y-x)^2/2} \, \frac{1}{2\sqrt \pi} e^{-(z-x)^2/2} \,$$

b) $$f(Y,Z)=\int_{-\infty}^{\infty}( \frac{1}{2\sqrt \pi} e^{-x^2/2} \, \frac{1}{2\sqrt \pi} e^{-(y-x)^2/2} \, \frac{1}{2\sqrt \pi} e^{-(z-x)^2/2} \, dx)$$

Answer 1

Part (a) work is correct, except the last line... there should be no integration. Part (b): integrate the answer for part (a) over $x$ only.