Difference between revisions of "Normal distribution"

Latest revision as of 01:31, 14 December 2017

Definition

The normal distribution has a Probability density function or PDF, $f:\mathbb{R}\rightarrow\mathbb{R}$ given by: $$\newcommand{\bigudot}{ \mathchoice{\mathop{\bigcup\mkern-15mu\cdot\mkern8mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}} }$$ $$\newcommand{\udot}{\cup\mkern-12.5mu\cdot\mkern6.25mu\!}$$ $$\require{AMScd}\newcommand{\d}[1][]{\mathrm{d}^{#1} }$$

• $$f(x):=\frac{1}{\sigma\sqrt{2\pi} } e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma} \right)^2}$$

The Cumulative density function or CDF is naturally given by:

• $$F(x):=P(-\infty < X < t)=\frac{1}{\sigma\sqrt{2\pi} }\int^t_\infty e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma} \right)^2}\d x$$

In this definition:

• $\sigma$ is the standard deviation of the distribution (so $\sigma^2$ is the variance) and
• $\mu$ is the mean

Notes:

The MDM of $X\sim\text{Nor}(0,\sigma^2)$ is $$\sqrt{\frac{2\sigma^2}{\pi} }$$ ^[1] , so is related the standard deviation linearly. It's also unaffected by the mean of the distribution - this hasn't been proved but is "obvious" and also verified experimentally.

References

1. Jump up ↑ From a friend's memory. It has been experimentally confirmed though and is at the very worst an extremely close approximation (on the order of $10^{-10}$)