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Normal Distribution

Also Gaussian Distribution · Gaussian · Gauss Distribution · Normally Distributed · Standard Normal Distribution · Normalverteilung · Gauß-Verteilung

A random variable X is normally distributed, X ~ N(μ, σ²), if its PDF is p(x) = exp(−(x − μ)²/(2σ²)) / √(2πσ²); its mean is μ and its variance σ².

Know these firstProbability Density Function Expected Value Variance

Properties#

Normalized: $\int_{-\infty}^{\infty} p(x)\,\mathrm{d}x = 1$ .
Parameters are the moments: the mean is $\mu_X = \mu$ and the variance is $\sigma_X^2 = \sigma^2$ . So the distribution is completely determined by its mean and variance.
Shape: symmetric bell curve around $\mu$ , maximum $1/\sqrt{2\pi\sigma^2}$ at $x = \mu$ , width set by $\sigma$ .
Standard normal: $\mathcal{N}(0, 1)$ . If $X \sim \mathcal{N}(\mu, \sigma^2)$ , then $(X - \mu)/\sigma \sim \mathcal{N}(0, 1)$ .
Vector case: the Multivariate Normal Distribution $\mathcal{N}(\mu, \Sigma)$ replaces $\sigma^2$ by a covariance matrix $\Sigma$ ; for $n = 1$ it reduces to this PDF.
Why it is everywhere: for many measurements, estimates such as the least squares estimate are approximately normally distributed even if the measurement errors are not (MSI script §4, p. 38).
Named after Carl Friedrich Gauss (1777–1855).

Example#

Sources#

MSI script p. 15 (§2.2.3), MSI Slides 02 > 2.2.3 The Normal Distribution (not published)
MSI script p. 38 (LS estimates are approximately normal)

Connections

Mentioned in inline links6

VL02 – Probability and Statistics in a NutshellMSI
The Normal Distribution $\mathcal{N}(\mu, \sigma^2)$ is named after its two parameters, which turn out to be exactly its mean and variance (Script p. 15). §
Normal Distribution: $\mathcal{N}(\mu, \sigma^2)$ , parameters are mean and variance §
Correlation
$\rho$ measures how well the best straight line fits the cloud, not how strongly $Z$ depends on $Y$ . In all three panels $Z$ is fully determined by $Y \sim \mathcal{N}(0, 1)$ (Normal Distribution), yet $\rho$ falls from the line (… §
Covariance
$N = 400$ samples of $(Y, Z)$ from a bivariate normal distribution with $\sigma_Y = \sigma_Z = 1$ , so $\sigma(Y, Z) = \rho$ (Correlation). Red points have a positive offset product, blue ones a negative. The sample covariance… §
Law of Large Numbers
Precise versions for i.i.d. $y_k$ with finite mean: the weak law gives convergence in probability, $P(|\frac1N\sum_k y_k - \mathbb{E}\{y\}| > \epsilon) \to 0$ for all $\epsilon > 0$ ; the strong law gives convergence with probability one. I… §
Multivariate Normal Distribution
Scalar case: for $n = 1$ , $\Sigma = \sigma^2$ and $\det(2\pi\sigma^2) = 2\pi\sigma^2$ , which gives the scalar Normal Distribution. §
Random Variables and Distributions
A random variable models a quantity that is not known before the experiment, such as a noisy measurement. It is described by probabilities of events, or more conveniently by a PDF. Joint, marginal and conditional densities describe several… §
Normal Distribution — the Gaussian, fixed by mean and variance §

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