Normal Distributions

The Standard Normal Distribution

Definition: Standard Normal Distribution

We say that a continuous random variable $X$ has the standard normal distribution if its probability density function $f$ can be written in terms of the real exponential function as follows:

$$f(x)=\frac{1}{\sqrt{2\pi }}{\mathrm{e}}^{-x^2/2}$$

NOTATION

$$X\sim \mathcal{N}(0,1)X\in \mathcal{N}(0,1)$$

In this case, it is also typical to denote the probability density function of $X$ by $\phi$ and its cumulative distribution function by $\Phi$.

Properties

Theorem: Expectation of the Standard Normal Distribution

The Expectation of a continuous random variable which has the standard normal distribution is zero.

$$X\sim \mathcal{N}(0,1)⟹\mathbb{E}[X]=0$$

PROOF

TODO

Theorem: Variance of the Standard Normal Distribution

The variance of a continuous random variable which has the standard normal distribution is one.

$$X\sim \mathcal{N}(0,1)⟹\mathrm{Var}(X)=1$$

PROOF

TODO

General Normal Distributions

Definition: Normal Distribution

We say that a continuous random variable $X$ has a normal distribution if there exist $\mu \in \mathbb{R}$, $\sigma >0$ and a continuous random variable $Z$ which has the standard normal distribution such that

$$X=\mu +\sigma Z.$$

NOTATION

$$X\sim N(\mu ,{\sigma }^2)X\in N(\mu ,{\sigma }^2)$$

Definition: Standardization

We call $Z$ the standardization of $X$.

Properties

Theorem: Cumulative Distribution Functions of Normal Distributions

The cumulative distribution function $F$ of a continuous random variable $X$ which has the normal distribution $\mathcal{N}(\mu ,{\sigma }^2)$ is

$$F(x)=\Phi \left(\frac{x-\mu }{\sigma }\right),$$

where $\Phi$ is the cumulative distribution function of $X$‘s standardization.

PROOF

TODO

Theorem: Probability Density of Normal Distributions

The probability density function $f$ of a continuous random variable $X$ which has the normal distribution $\mathcal{N}(\mu ,{\sigma }^2)$ is

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

where ${\mathrm{e}}^{-\frac{1}{2}{\left(\frac{x-\mu }{\sigma }\right)}^2}$ is the real exponential function and $\phi$ is the probability density function of $X$‘s standardization.

PROOF

TODO

Theorem: Expectation of Normal Distributions

The Expectation of a continuous random variable $X$ which has the normal distribution $\mathcal{N}(\mu ,{\sigma }^2)$ is $\mu$.

$$X\sim \mathcal{N}(\mu ,{\sigma }^2)⟹\mathbb{E}(X)=\mu$$

PROOF

TODO

Theorem: Variance of Normal Distributions

The variance of a continuous random variable $X$ which has the normal distribution $\mathcal{N}(\mu ,{\sigma }^2)$ is ${\sigma }^2$.

$$X\sim \mathcal{N}(\mu ,{\sigma }^2)⟹\mathrm{Var}(X)={\sigma }^2$$

PROOF

TODO

Theorem: The 68–95–99.7 Rule

If a continuous random variable $X$ has the normal distribution $\mathcal{N}(\mu ,{\sigma }^2)$, then

$$\begin{array}{r}P(|X-\mu |<1\sigma )\approx 68.27\%\\ P(|X-\mu |<2\sigma )\approx 95.45\%\\ P(|X-\mu |<3\sigma )\approx 99.73\%\end{array}$$

PROOF

TODO