Normal Distribution

Gist

The most famous of the distributions. It's everywhere!

Mathematics

A continuous probability density distribution:

\frac{1}{\sqrt{2 π σ^{2}}} e^{- \frac{(x - μ)^{2}}{2 σ^{2}}} .

Note, the multiplier ( $\frac{1}{\sqrt{2 π σ^{2}}}$ ) is simply to normalize the Probability Density (Mass) Function (see "How was Normal Distribution Derived" in the References).

And for the cumulative probability function it is:

Φ (x) = \frac{1}{\sqrt{2 π}} \int_{- \infty}^{x} e^{\frac{- t^{2}}{2}} d t

Properties

Mean: $μ$
Variance: $σ^{2}$
Standard deviation: $σ$

Rule

About 68% of values fall within one standard deviation of the mean, about 95% within two, and about 99.7% within three. This is sometimes called the 68-95-99.7 rule.

Relationship

With the Gamma Distribution, as you increase the shape parameter, the Gamma distribution approaches normal (important for simulating wait times)
It can be an approximation to a Binomial Distribution
For Beta Distribution, when the shape parameters are equal, thd distribution is normal shaped

Flavors

Standard normal distribution is when the $μ = 0$ and $σ = 1$ . You can convert any normal distribution to a standard normal distribution using z-score

Uses

References

StackExchange - How was the Normal Distribution Derived