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Let $Z_1, Z_2 \sim \mathcal{N}(0,1)$ be i.i.d random variables. I wish to find the distribution of \begin{align} \frac{Z_1^2 + Z_2^2}{Z_1+Z_2} \,. \end{align} It is well known that $W = Z_1^2 + Z_2^2 \sim \chi^2_2$ and $Q = Z_1 + Z_2 \sim N(0,2)$. If $W$ and $Q$ were independent, then we might hope to invert this result relating to $t$-distributions, but of course they are not independent since \begin{align} Q^2 = W + 2Z_1 Z_2 \,. \end{align} Any thoughts on how to proceed are welcome!

lemmykc
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  • It looks bimodal and heavy-tailed, vaguely reminding me of the reciprocal standard normal distribution though not as extreme – Henry Dec 01 '21 at 02:27
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    A random variable which functionally depends from another one is not necessarily stochastly dependent each other. – Davi Américo Dec 01 '21 at 03:35
  • @DaviAmérico Fair point. It looks like [this post](https://stats.stackexchange.com/questions/429757/finding-joint-distribution-of-x-y-x2-y2-where-x-y-are-independent-s?rq=1) may be of use, though I still need to work through it to check for independence. – lemmykc Dec 01 '21 at 05:50

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One way to proceed is to transform to polar coordinates $(Z_1,Z_2) \mapsto (R,\Theta)$ such that

$$Z_1=R\cos\Theta \quad, \quad Z_2=R\sin\Theta$$

Then,

$$\frac{Z_1^2+Z_2^2}{Z_1+Z_2}=\frac{R}{\cos\Theta+\sin\Theta}=\frac{R}{\sqrt 2\sin\left(\Theta+\frac{\pi}4\right)}$$

Now $R$ and $\Theta$ are independently distributed with $R$ having a Rayleigh distribution and $\Theta$ having a uniform distribution on $(0,2\pi)$. Independence of $R$ and $\Theta$ implies the independence of $R$ and $\sin\left(\Theta+\frac{\pi}4\right)$. So it is theoretically possible to derive this distribution.

StubbornAtom
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    Use $$\frac{Z_1^2+Z_2^2}{Z_1+Z_2} = \frac{1}{\sqrt 2}\left[\frac{X^2+Y^2}{X}\right]$$ where the *i.i.d.* assumption implies $X=(Z_1+Z_2)/\sqrt 2$ and $Y=(Z_1-Z_2)/\sqrt 2$ are independent standard Normal variables. This will (a) simplify the polar coordinate expression and (b) point out a missing factor of $\sqrt{2}$ in the analysis. Equivalently, let $(Z_1,Z_2) = R\sqrt{2}(\cos(\theta-\pi/4), \sin(\theta-\pi/4)).$ – whuber Dec 01 '21 at 15:45
  • @whuber I agree the expression looks better and simpler with that transformation. But does that make the problem easier to solve? – StubbornAtom Dec 01 '21 at 16:28
  • Yes, because (1) no trigonometric manipulation is needed and (2) it automatically avoids a potential error in overlooking the factor of $1/\sqrt 2$ that would bite the casual reader. – whuber Dec 01 '21 at 16:33
  • @whuber Any advantage apart from that? The remaining job looks identical to what I presently have. – StubbornAtom Dec 01 '21 at 16:51
  • Correct: that's why I offer this only as a comment, not as a separate answer. – whuber Dec 01 '21 at 17:05