Let $X, Y, Z \sim N(0, 1)$ and are independent. How to show that $\frac{X + ZY}{\sqrt{1 + Z^2}} \sim N(0, 1)$?

Question

Let $X, Y, Z$ be i.i.d. random variables distributed as $N(0, 1)$. How to show that $\dfrac{X + ZY}{\sqrt{1 + Z^2}} \sim N(0, 1)$?

The only solution I see is to progressively find the distributions of

1. First, $ZY$
2. Then $U := X + ZY$
3. Then $V := \sqrt{1 + Z^2}$
4. Then $\dfrac{U}{V}$

But finding these distributions appears to be a nightmare in terms of the number of calculations to be done.

And this solution doesn't really explain the intuition behind this problem: how could this expression randomly end up being $N(0, 1)$-distributed? That makes me wonder if there is a more nice solution. I also think, this is probably a well-known problem, but I couldn't find a solution anywhere.

Any help would be greatly appreciated!

Answer 1

As you did in this question, it suffices to define the random variable $\alpha$ such that $$\cos \alpha =\frac{1}{\sqrt{1+Z^2}}$$ (we don't need to know its distribution) and calculate the characteristic distribution.

Or a more direct approach is to calculate the characteristic function with the conditional expectation $E( ...|Z)$ on the random variable $Z$. You'll have the same result.

Answer 2

If $X,Y\sim N(0,1)$ then the rv $a(X+bY)$ is $ N(0,a^2(1+b^2))$. In particular, if $a=1/\sqrt{1+b^2}$, then it's $N(0,1)$.

Then, letting $U=\frac{X + ZY}{\sqrt{1 + Z^2}}$, we know $U | Z \sim N(0,1)$ : it does not depend on $Z$).

Hence $f_U=f_{U|Z}$ ($U$ and $Z$ are independent) and $U \sim N(0,1)$