Normal Distribution
Visualization
Math
The formula of normal distribution is
where $\mu$ controls the “center” or “peak” of the distribution and $\sigma$ tells us how “wide” or “disperse” the distribution is.
To understand the distribution, we take some limits.
$x = \mu$
First of all, when $x = \mu$ we have
Notice the argument of the exponential is some squared value and can not be negative. This condition gives us the peak value.
$x=\mu-a$ and $x=\mu + a$
For $x=\mu-a$, we have
For $x=\mu + a$, we have
which is exactly the same as the previous case.
The distribution is symmetric around $x=\mu$.
$x=\pm \infty$
We have 0 for both cases.
Tricks
Integral
We integrate distributions a lot. For Gaussian distribution, it is quite helpful to remember the following identity.
It tells us that for $\mu=0$ and $\sigma=1/\sqrt{2}$, the area under the distribution is $\sqrt{\pi}$.
Hey, it is time to ask the question. Where the hell is the circle?
Error Function
The error function is defined as
Obviously, the coefficient $\frac{1}{\sqrt{\pi}}$ normalizes the function to be within $[-1,1]$.