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

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

Uses

References

StackExchange - How was the Normal Distribution Derived