### Deriving the normal distribution

In every introductory statistics class, we learned about the normal distribution, which has Probability Density Function (PDF):

$$\tag{1} f(x | \mu, \sigma^2) = \frac{1}{\sqrt{2\pi \sigma^2}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}$$

This looks like a fairly complicated equation, but the resulting graph (shown above) has some very cool properties (integrates to 1, represents real-valued random variables whose distributions are not known etc…). I’ve always wondered how this is derived, and I finally found some answers via great videos and online forums. I will give an overview of the derivation here, based on YouTuber Mathoma’s amazing video (linked above).

Part 1: The Theory

Mathoma gave a great analogy about how to understand this distribution: imagine you are throwing darts on a polar coordinate system, with the goal of hitting the center $(0,0)$. Now, given an arbitrary dart landing on coordinate $(r, \theta)$, we can also say that the coordinate is $(x, y)$ if we convert from polar to cartesian.

We have to make a couple assumptions here before moving forward. First, we assume that $x$ and $y$ are statistically independent. Second, we assume that the PDF is rotationally invariant, which means the distribution of where my dart lands only depends on the distance $r$, of the dart to the center.

With those assumptions, I can define PDF $\varphi(r) = f(x)f(y)$. This can be rewritten as

\begin{align} \tag{2} \varphi(\sqrt{x^2 + y^2}) &= f(x)f(y) \end{align}

Next, suppose $y=0$. We will then have

\begin{aligned} \varphi(\sqrt{x^2 + 0^2}) &= f(x)f(0) \\ \varphi(x) &= f(x)\lambda, \text{ where \lambda is a constant} \end{aligned}

Plugging this back into Eq 2, we have

\begin{align} \tag{3} \lambda f(\sqrt{x^2 + y^2}) = f(x)f(y) \end{align}

Next, we will determine the expression for $f(x)$. First, we rewrite Eq 3 as

\begin{aligned} \frac{\lambda f(\sqrt{x^2 + y^2})}{\lambda^2} = \frac{f(x)}{\lambda} \frac{f(y)}{\lambda} \end{aligned}

For simplicity in analyzing the equation, define $g(x) = \frac{f(x)}{\lambda}$. We now have

\begin{align} \tag{4} g(x)g(y) = g(\sqrt{x^2 + y^2}) \end{align}

What kind of function should $g$ be so that Eq 4 is valid? Upon some inspection, we can see that $g$ should be an exponential function. Example: suppose we have $h(x) = e^x$, then $h(x)h(y) = e^xe^y = e^{x+y} = h(x+y)$.

Similarly, let $g(x) = e^{Ax^2}$, where $A$ is a constant.

\begin{align*} g(x)g(y) &= e^{Ax^2}e^{Ay^2}\\ &= e^{A(x^2 + y^2)}\\ &= e^{A(\sqrt{x^2 + y^2})^2}\\ &= g(\sqrt{x^2 + y^2}) \end{align*}

In turn, our PDF $f$ should be

\begin{align} \tag{5} f(x) = \lambda e^{Ax^2} \end{align}

Plotting this equation out with $A=-1$ (more on why $A$ is negative later) and $\lambda=1$, we see that it takes a gaussian form!

Part 2: Massaging the Equation

The remainder of this derivation serves to massage Eq 5 into the class of gaussians we are interested in, the normal gaussian.

First, we introduce a constraint on the function: since we are modeling probability, it makes sense for $f(x)$ to integrate to $1$.

\begin{aligned} \int_{-\infty}^{\infty}f(x)dx = 1 \end{aligned}

Instead of using constant $A$, we will set $A = -h^2$, where $h$ is a constant variable. There are several reasons for this: First, it makes sense for $A$ to be negative, because we want this function (which models the probability) to decrease as we move to $+\infty$. Second, the $-h^2$ form will help when we do the integration.

We determine the value of $h^2$:

\begin{aligned} \int_{-\infty}^{\infty}\lambda e^{-h^2x^2}dx &= 1\\ \lambda\int_{-\infty}^{\infty}e^{-h^2x^2}dx &= 1\\ \end{aligned}

Perform u-substitution, with $u = hx$, $du = hdx$, and $dx = \frac{1}{h}du$. We now get

\begin{align} \tag{6} \frac{\lambda}{h}\int_{-\infty}^{\infty}e^{-u^2}du = 1 \end{align}

Interestingly, the integral in Eq 6 is actually famous (it has a name!). The Gaussian integral, also known as the Euler-Poisson integral, is equal to $\sqrt{\pi}$ (refer to link for the integral computation).

We can now compute $h^2$:

\begin{aligned} \frac{\lambda}{h}\sqrt{\pi} &= 1\\ h &= \lambda \sqrt{\pi}\\ h^2 &= \lambda^2\pi \end{aligned}

Eq 5 becomes

\begin{align} \tag{7} f(x) = \lambda e^{-\pi \lambda^2x^2} \end{align}

If we plot $f(x)$ using different $\lambda$ values, we see that as $\lambda$ increases, the variance $\sigma^2$ decreases, since more of the area is accumulated at $x=0$. See plots for $\lambda=1$, $\lambda=2$, $\lambda=3$ as examples.

Next, we need to find the relationship between $\lambda$ and variance $\sigma^2$. From definition of variance, we see that $Var(X) = E[(X - \mu)^2]$. In our case, $\mu$ is 0, so we have:

\begin{aligned} Var(x) = \sigma^2 &= \int_{-\infty}^{\infty}x^2 \lambda e^{-\pi \lambda^2x^2}dx \\ &= \lambda \int_{-\infty}^{\infty} x \cdot x \cdot e^{-\pi \lambda^2x^2}dx \end{aligned}

We will evaluate this integral via integration by parts. Recall that

\begin{aligned} \int udv = uv - \int vdu \end{aligned}

Let $u = x$, $du = dx$, $dv = xe^{-\pi \lambda^2x^2}dx$, $v = -\frac{1}{2\pi\lambda^2}e^{-\pi \lambda^2x^2}$:

\begin{aligned} Var(x) = \sigma^2 &= \lambda \left(x\left(-\frac{1}{2\pi\lambda^2}e^{-\pi \lambda^2x^2}\Biggr|_{-\infty}^{\infty}\right) - \int_{-\infty}^{\infty}\left(-\frac{1}{2\pi\lambda^2}e^{-\pi \lambda^2x^2}\right)dx\right)\\ &= \lambda \left((0) + \int_{-\infty}^{\infty}\frac{1}{2\pi\lambda^2}e^{-\pi \lambda^2x^2}dx\right)\\ &= \frac{1}{2\pi\lambda^2}\int_{-\infty}^{\infty}\lambda e^{-\pi \lambda^2x^2}dx \end{aligned}

The reason we switched the position of $\lambda$ and $\frac{1}{2\pi\lambda^2}$ is so we can massage the integral to be the form of the gaussian PDF, which we know integrates to 1. Now, with the equation simplified, we can solve for $\lambda$ in terms of $\sigma^2$.

\begin{aligned} \sigma^2 &= \frac{1}{2\pi\lambda^2}\\ \lambda^2 &= \frac{1}{\sigma^2 2\pi}\\ \lambda &= \frac{1}{\sqrt{2\pi\sigma^2}} \end{aligned}

$\lambda$ and $\sigma$ are inversely proportional, as expected. We can plug in our result here back into Eq 7:

\begin{align} \tag{8} f(x) &= \frac{1}{\sqrt{2\pi\sigma^2}}e^{-\pi \left(\frac{1}{(\sqrt{2\pi\sigma^2})^2}\right)x^2} \\ &= \frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{x^2}{2\sigma^2}}\nonumber \end{align}

We are almost done. The above equation has mean $\mu = 0$, but if we want to represent $f(x|\mu, \sigma^2)$, we need to add in $f(x-\mu)$. Therefore, our general PDF equation for the normal distribution is:

\begin{align} \tag{9} f(x | \mu, \sigma^2) = \frac{1}{\sqrt{2\pi \sigma^2}}e^{-\frac{(x-\mu)^2}{2\sigma^2}} \end{align}

Eq 9 matches Eq 1, and we are now done $\blacksquare$