Bell Curve Integral

Evaluating this integral requires using two nonstandard tricks (i.e. things not taught in calc 1).

evaluating the product of the integral and itself
using a nonlinear, 2D transformation of variables (actually just converting to polar coordinates).

This can also be done in 2D cartesian coordinates, by using x and y, and the substitution $$y=sx$$ (and $$dy=sdx$$). This is just another trick for bringing one of the variables of integration down from the exponent to the coefficient, allowing easy integration of $$e^FdF$$.

Area under $$e^{-z^2}$$

The bell curve integral:

$$S = \int_{-\infty}^{\infty} e^{-z^2}dz$$

Let's compute the square of S instead:

$$S^2 = \int_{-\infty}^{\infty} e^{-z^2}dz \cdot \int_{-\infty}^{\infty} e^{-z^2}dz$$

Swap out a dummy variable:

$$ = \int_{-\infty}^{\infty} e^{-x^2}dx \cdot \int_{-\infty}^{\infty} e^{-y^2}dy$$

Combine integrals:

$$ = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} e^{-(x^2+y^2)}dxdy$$

Convert from cartesian to polar coordinates, $$dA = dxdy = rdrd\theta$$

$$ = \int_{0}^{2\pi} \int_{0}^{\infty} e^{-r^2}r dr d\theta$$

Integrand doesn't depend on $$\theta$$:

$$ = 2\pi \int_{0}^{\infty} e^{-r^2}r dr$$

substitute $$s=r^2$$, $$2rdr=ds$$:

$$ = \pi \int_{0}^{\infty} e^{-s}ds$$

$$S^2 = \pi$$

$$S = \sqrt{\pi} \blacksquare$$

(ick, not sure how to right-align the tombstone symbol)

Shift

$$S = \int e^{-(x-x_0)^2} dx$$

simply substitute $$y=x-x_0$$, $$dy=dx$$, giving

$$S = \int e^{-y^2} dy = \sqrt{\pi} \blacksquare$$

Scale

$$S = \int e^{-ax^2} dx = \int e^{-(\sqrt{a}x)^2} dx$$

let $$y = \sqrt{a}x$$, or $$y^2 = ax^2$$, $$dx = dy/\sqrt{a}$$

$$S = \frac{1}{\sqrt{a}}\int e^{-y^2} dy$$

$$S = \sqrt{\frac{\pi}{a}} \blacksquare$$

Or, for slightly different notation,

$$S = \int e^{-(ax)^2} dx$$

let y = ax, or y^2 = (ax)^2, dx = dy/a

$$S = \frac{1}{a}\int e^{-y^2} dy$$

$$S = \sqrt{\frac{\pi}{a^2}} = \frac{\sqrt{\pi}}{a} \blacksquare$$

This shows that the coefficient for the normal distribution, $$e^{-\frac{(x-\mu)^2}{2\sigma^2}}$$, should be the inverse of $$\sqrt{\pi \cdot 2\sigma^2}$$, or $$\frac{1}{\sqrt{\pi \cdot 2\sigma^2}}$$

General quadratic in exponent

This comes about for, e.g., computing matched filter response for bell curve $$(\int e^{-a\tau^2}e^{-a(t-\tau)^2}d\tau)$$, or finding the probability distribution of a sum of Gaussian random variables. We can ignore the constant term, since it's just a multiplicative factor.

$$S = \int e^{-(ax^2 + 2bx)}dx$$

To complete the square, substitute $$y=\sqrt{a}x$$:

$$S = \frac{1}{\sqrt{a}}\int e^{-\left(y^2 + \frac{2b}{\sqrt{a}}y\right)}dy$$

multiply by $$1 = e^{-b^2/a}e^{b^2/a}$$ (add constant term to the quadratic exponent):

$$S = \frac{1}{\sqrt{a}} e^{\frac{b^2}{a}}\int e^{-\left(y^2 + \frac{2b}{\sqrt{a}}y + \frac{b^2}{a}\right)}dy$$

Factor:

$$S = \frac{1}{\sqrt{a}} e^{\frac{b^2}{a}}\int e^{-\left(y + \frac{b}{\sqrt{a}}\right)^2}dy$$

Substitute $$z = y+b/\sqrt{a}$$ to clarify:

$$S = \frac{1}{\sqrt{a}} e^{\frac{b^2}{a}}\int e^{-z^2}dz$$

$$S = \sqrt{\frac{\pi}{a}} e^{\frac{b^2}{a}}$$

Summary

$$S = \int e^{-(ax^2 + bx)}dx$$

$$ = \int e^{-\left((\sqrt{a}x)^2+\frac{b}{\sqrt{a}}(\sqrt{a}x)+\frac{b^2}{4a}\right)}e^{\frac{b^2}{4a}}dx$$

which suggests

$$S = e^{\frac{b^2}{4a}}\int e^{-\left(\sqrt{a}x + \frac{b}{2\sqrt{a}}\right)^2}dx$$

$$ = e^{\frac{b^2}{4a}}\int e^{-(\sqrt{a}u)^2}du = \frac{e^{\frac{b^2}{4a}}}{\sqrt{a}} \int e^{-v^2}dv$$

or, with slightly different notation: