# Are there two distributions whose product equals a gaussian?

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?

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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.

winperikle
July 12, 2019 10:15 AM