Probability theory and stochastic processes Definition

Density of the normal distribution

Definition

Let $\mu \in \mathbb{R}$ and $\sigma^2 > 0$ . The density of the normal distribution with mean $\mu$ and variance $\sigma^2$ is the function $\varphi_{\mu, \sigma^2} \colon \mathbb{R} \to (0, \infty)$ with $\varphi_{\mu, \sigma^2}(x) = \frac{1}{\sqrt{2\pi\sigma^2}} \exp\left(-\frac{(x - \mu)^2}{2\sigma^2}\right).$ The standard normal density is $\varphi(x) = \varphi_{0, 1}(x) = \frac{1}{\sqrt{2\pi}} e^{-x^2/2}$ .

Remarks

The integral of $\varphi_{\mu, \sigma^2}$ over $\mathbb{R}$ is $1$ . This follows from the Gaussian integral $\int_{-\infty}^{\infty} e^{-x^2} \, dx = \sqrt{\pi}$ . So $\varphi_{\mu, \sigma^2}$ is a probability density function, the density of the normal distribution $\mathcal{N}(\mu, \sigma^2)$ . The function is symmetric about $\mu$ , has its maximum $1/\sqrt{2\pi\sigma^2}$ at $x = \mu$ , and has inflection points at $\mu \pm \sigma$ . It satisfies $\varphi_{\mu, \sigma^2}(x) = \frac{1}{\sigma} \varphi\left(\frac{x - \mu}{\sigma}\right)$ with $\sigma = \sqrt{\sigma^2}$ .

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Density of the normal distribution definition

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Tag	`027JF1FBB0`
URI	`m2://027JF1FBB0`
Kind	definition
Writing	W0 stub
MSC	60E05
Defines	Density of the normal distribution
Uses	Probability density function

Density of the normal distribution

Definition

Remarks

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