It is said that we may use the binomial coefficients ( a layer from Pascal's triangle) to approximate the 1-D Gaussian kernel with certain $\sigma$, where $\frac{n}{4} = \sigma$ and $n$ is the index of the layer.

This works really nice when we want to generate a gaussian quickly. But is there a way to prove this? Or just coincidence?

And how do we decide the variance of certain layer of Pascal's triangle? Say the third layer is [1 3 3 1] (consider [1 1] as the first layer), and it is used to approximate gaussian kernel of $\sigma = \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2}$. But how do we prove this?

  • $\begingroup$ maybe should ask the math SE this question. we be just dumb electrical engineers. we don't know which end of the integral symbol is up. $\endgroup$ – robert bristow-johnson Mar 6 '18 at 8:56

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