by Hilder Vítor Lima Pereira
Last Updated July 12, 2019 10:19 AM - source

Are there two distributions $X$ and $Y$ over $\mathbb{R}$ such that the distribution of the product $XY$ follows a Gaussian distribution?

If $X$ and $Y$ are both standard normal then $X^2 + Y^2 \sim \chi^2_2 = \mathcal{E}(1/2)$ and the angle of $(X,Y)$ in the plane whose sinus is given by $\frac{Y}{\sqrt{X^2 + Y^2}}$ is $\mathcal{U}_{[-\pi,\pi]}$.

Thus let $\theta \sim \mathcal{U}_{[-\pi,\pi]}$ and $Z \sim \mathcal{E}(1/2)$, then

$$ Y = \sqrt{Z} \text{sin}(\theta) \sim \mathcal{N}(0,1) $$

where $\sqrt{Z}$ follows a Rayleigh distribution with scale 1 while $\text{sin}(\theta)$ follows the Arcsine distribution.

This method is known as the Box-Muller transform which enables one to generate standard normal variable from independent uniform variables.

July 12, 2019 10:15 AM

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