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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

\[ X \sim \mathcal{N}(0,1) \qquad X \in \mathcal{N}(0,1) \]

In this case, it is also typical to denote the probability density function of \(X\) by \(\varphi\) 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.

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

TODO

Theorem: Variance of the Standard Normal Distribution

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

\[ X \sim \mathcal{N}(0,1) \implies \operatorname{Var}(X) = 1 \]
Proof

TODO

General Normal Distributions#

Definition: Normal Distribution

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

\[ X = \mu + \sigma Z. \]

Notation

\[ X \sim N(\mu, \sigma^2) \qquad X \in N(\mu, \sigma^2) \]

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

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

where \(\mathrm{e}^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}\) is the real exponential function and \(\varphi\) 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 \(\mu\).

\[ X \sim \mathcal{N}(\mu, \sigma^2) \implies \mathbb{E}(X) = \mu \]
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 \(\sigma^2\).

\[ X \sim \mathcal{N}(\mu, \sigma^2) \implies \operatorname{Var}(X) = \sigma^2 \]
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

\[ \begin{aligned} P(|X - \mu| \lt 1\sigma) \approx 68.27 \% \\ P(|X - \mu| \lt 2\sigma) \approx 95.45 \% \\ P(|X - \mu| \lt 3\sigma) \approx 99.73 \% \end{aligned} \]
Proof

TODO