[Math] Formula for skewed bell curve

normal distribution

I can plot a bell curve using the formula:
$$
Y = \frac{1}{\sqrt{2\pi S^2} } e^{ -\frac{(X-A)^2}{(2S)^2} }
$$
where A=mean and S=standard deviation. This is obviously on an X,Y plot.

I want to add a skew value of -1 to 1 where 0 means no skew (a normal distribution). Is there a commonly accepted formula that will skew the bell curve?

Best Answer

There is a simple, well-known modification to skew a normal distribution as follows,

$$Y_k(X) = Y(X) + k\frac{dY(X)}{dX}$$

where $k$ is the skew parameter to regulate the amount of skews and

$$\frac{dY(X)}{dX}= \frac{1}{\sqrt{2\pi S^2} } e^{ -\frac{(X-A)^2}{(2S)^2} }\left(-\frac{X-A}{2S^2}\right) $$

This is often used in financial modeling, and it preserves the unit cumulative integration.

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