Normal Distribution

Statements

Definition. Let $X$ be a random variable with outcome set $\mathbb{R}$ and PDF $f(x)=\frac{1}{\sqrt{2\pi \sigma }}\exp \left(-\frac{(x-\mu {)}^2}{2{\sigma }^2}\right),$ Then $X$ is called a Gaussian random variable and said to be distributed according to a Gaussian distribution with parameters $\mu \in \mathbb{R}$ and ${\sigma }^2>0$, for which we write $X\sim \mathcal{N}(\mu ,{\sigma }^2)$. These parameters are the expectation and standard deviation, respectively.

The normal distribution is said to be a standard normal distribution, when $\sigma =1$ and $\mu =0$.

Plot

For the normal distribution, the values less than one standard deviation away from the mean account for 68.27% of the set; while two standard deviations from the sample mean account for 95.45%; and three standard deviations account for 99.73%.

References

Ost20, Definition 7.0.1

[Normal distribution - Wikipedia](<https://en.wikipedia.org/wiki/Normal_distribution)