# Gaussian random variable (Gaussian RV)

*Entry of the [Dictionary of Applied Machine Learning](https://dictionaryofml.org) (course edition), by Alexander Jung, Aalto University. CC BY 4.0. Updated 2026-10-29. [HTML](https://dictionaryofml.org/terms/gaussrv.html) · [typeset PDF](https://dictionaryofml.org/terms/gaussrv.pdf) · [LaTeX](https://dictionaryofml.org/terms/gaussrv.tex) · [Python demo](https://dictionaryofml.org/terms/gaussrv.py) · [notebook](https://dictionaryofml.org/terms/gaussrv.ipynb)*

Math is LaTeX with the book's macros expanded, so no notation package is needed to read it.

## Abstract

A Gaussian random variable (Gaussian RV) is a real-valued random variable whose probability density function is proportional to the exponential of a negative quadratic. The probability density function is determined by two numbers, the mean $\mu$ and the variance $\sigma^{2}$ of the random variable; the probability distribution with these two parameters is denoted by $\mathcal{N}\left(\mu,\sigma^{2}\right)$. The standard Gaussian random variable is the special case $\mu = 0$ and $\sigma^{2} = 1$; a general Gaussian random variable is obtained from it by scaling and shifting. A Gaussian random vector is an affine transformation of independent standard Gaussian RVs. Gaussian RVs arise as limits of averages of many independent RVs through the central limit theorem. Among all random vectors with a given covariance matrix, the Gaussian random vector maximizes the differential entropy.

## Definition

The measurement error of a sensor, such as a thermometer, is usually modeled as a Gaussian random variable. A real-valued random variable $z$ is a standard Gaussian random variable if its probability density function is (Bertsekas and Tsitsiklis, 2008; Gray, 2009; Papoulis and Pillai, 2002)

$$
p^{(z)}\left(\eta\right) = \frac{1}{\sqrt{2\pi}} \exp\,(-\eta^2/2) \text{.}
$$

Given a standard Gaussian random variable $z$, a general Gaussian random variable $x$ with mean $\mu$ and variance $\sigma^2$ can be constructed via $x := \sigma z + \mu$, so that the probability distribution of $x$ is the normal distribution $\mathcal{N}\left(\mu,\sigma^{2}\right)$, denoted by $x \sim \mathcal{N}\left(\mu,\sigma^{2}\right)$. The probability density function of $x$ is, for every $\eta \in \mathbb{R}$,

$$
p^{(x)}\left(\eta\right) = \frac{1}{\sqrt{2\pi\sigma^{2}}}
\exp\Big( -\frac{(\eta-\mu)^{2}}{2\sigma^{2}} \Big) \text{.}
$$

The standard Gaussian random variable is the special case $\mu = 0$ and $\sigma^{2} = 1$.

The two parameters of $\mathcal{N}\left(\mu,\sigma^{2}\right)$ act separately on the probability density function, as shown in Fig. 1: the parameter $\mu$ is the mean of $x$, and the parameter $\sigma$ is its standard deviation, the square root of the variance $\sigma^{2}$. Adding $\mu$ to the scaled random variable $\sigma z$ shifts the probability density function along the horizontal axis by $\mu$, without changing its shape. For $\sigma>0$, multiplying $z$ by $\sigma$ stretches the probability density function horizontally by a factor of $\sigma$ and lowers its peak by the same factor, so the area under it remains $1$.

![Figure 1 of the entry gaussrv](https://dictionaryofml.org/terms/fig/gaussrv-fig1.svg)

*Figure 1: The solid curve is the probability density function of a standard Gaussian random variable $z$. The dashed curve is the probability density function of the Gaussian random variable $x = \sigma z + \mu$ with $\mu = 2$ and $\sigma = 1.5$. The shift parameter $\mu$ moves the mean from $0$ to $\mu$, while the scale parameter $\sigma$ changes the standard deviation from $1$ to $\sigma$ and the variance from $1$ to $\sigma^{2}$. A vertical ruler marks the mean of each curve, at $\eta = 0$ and at $\eta = \mu$, and each horizontal double arrow runs from the mean of its own curve to one standard deviation away, $1$ for the solid curve and $\sigma$ for the dashed one. The gray arrow from peak to peak marks the shift and the scaling that carry the one probability density function to the other*

Two ways to generate a Gaussian random variable from other sources of randomness are in common use. For exact generation, let $U$ be a random variable uniformly distributed on the open interval $(0,1) \subset \mathbb{R}$ and let $\Phi$ be the cumulative distribution function of a standard Gaussian random variable. Then $\mu + \sigma\Phi^{-1}(U) \sim \mathcal{N}\left(\mu,\sigma^{2}\right)$, the same probability distribution as $x$ but obtained from a different source of randomness.

For approximate generation, let $m$ be a positive integer and let $v^{(1)},\,\ldots,\,v^{(m)}$ be independent and identically distributed RVs, not necessarily Gaussian, with mean $0$ and finite, nonzero variance $\tau^{2}$. Then the normalized sum

$$
\frac{1}{\tau\sqrt{m}}\sum_{i=1}^{m}v^{(i)}
$$

converges in distribution to a standard Gaussian random variable as $m$ increases. Thus, the central limit theorem makes a Gaussian approximation appropriate for averages of many independent contributions, which is why the measurement error of a sensor is usually modeled as Gaussian, although a finite sum need not be Gaussian (Ross, 2014). Fig. 2 shows both halves of that statement for coin flips, each taking the values $+1$ and $-1$ with probability $1/2$. The probability mass function of the normalized sum of $m = 8$ of them already follows the shape of the standard Gaussian probability density function, and the sum still takes only $9$ values. Convergence in distribution is convergence of the cdfs, so the largest gap between the cumulative distribution function of the normalized sum and that of a standard Gaussian random variable measures how far the approximation still is. It is $0.137$ for $m = 8$ and $0.050$ for $m = 64$.

![Figure 2 of the entry gaussrv](https://dictionaryofml.org/terms/fig/gaussrv-fig2.svg)

*Figure 2: Here, $s$ is the sum of $m = 8$ independent and identically distributed RVs, each taking the values $+1$ and $-1$ with probability $1/2$ (coin flips). The stems carry the probability mass function of the normalized sum $s/\sqrt{m}$, which takes $9$ values spaced $2/\sqrt{m}$ apart, so the normalized sum is not Gaussian. The dashed curve is the probability a standard Gaussian random variable puts on the cell of that width around each value, its probability density function there times $2/\sqrt{m}$. Data generated by pythondemos/gaussrv.py*

A Gaussian random vector ${\mathbf x} \in \mathbb{R}^{d}$ with covariance matrix ${\mathbf C}$ and mean ${\boldsymbol \mu}$ can be constructed as (Gray, 2009; Lapidoth, 2017; Papoulis and Pillai, 2002)

$$
{\mathbf x} := {\mathbf A} {\mathbf z} + {\boldsymbol \mu} \text{.}
$$

Here, ${\mathbf z} := \big( z_{1}, \,\ldots, \,z_{d} \big)^{\top}$ is a vector of independent and identically distributed standard Gaussian RVs. The matrix ${\mathbf A} \in \mathbb{R}^{d \times d}$ is a matrix square root of ${\mathbf C}$, i.e., any matrix with ${\mathbf A} {\mathbf A}^{\top} = {\mathbf C}$ (a Cholesky factor, for instance). The probability distribution of a Gaussian random vector is referred to as the multivariate normal distribution, denoted by $\mathcal{N}\left({\boldsymbol \mu},{\mathbf C}\right)$. A Gaussian random vector need not have independent components; its components are independent exactly when its covariance matrix is diagonal.

A Gaussian random vector ${\mathbf x}=\big(x_{1},\,\ldots,\,x_{d}\big)^{\top}$ can be interpreted as a stochastic process indexed by the finite set $\{1,\,\ldots,\,d\}$. More generally, a Gaussian process is a stochastic process over an index set $\mathcal{I}$ whose restriction to every finite subset of $\mathcal{I}$ is a Gaussian random vector (Rasmussen and Williams, 2006).

A standard Gaussian random variable $z$ obeys the Gaussian tail bound $\mathbb{P}\left(|z| \geq t\right) \leq 2 \exp(-t^{2}/2)$ for every positive number $t$. That bound also controls functions of independent and identically distributed RVs. Consider a standard Gaussian random vector ${\mathbf z} \in \mathbb{R}^{d}$ and a function $f :
\mathbb{R}^{d} \to \mathbb{R}$ that is Lipschitz with constant $L$, i.e., $\big| f({\mathbf z}) - f({\mathbf z}') \big| \leq
L \Vert {{\mathbf z} - {\mathbf z}'} \Vert$ for every pair of vectors ${\mathbf z}, {\mathbf z}' \in \mathbb{R}^{d}$. The deviation of $f({\mathbf z})$ from its expectation is then a sub-Gaussian random variable with parameter at most $L$. For every positive number $t$,

$$
\mathbb{P}\left(\big| f({\mathbf z}) - \mathbb{E}\big\{ f({\mathbf z}) \big\} \big| \geq t\right)
\leq 2 \exp\Big( \frac{-t^{2}}{2L^{2}} \Big)
\text{.}
$$

The number $d$ of arguments does not appear in this concentration inequality (Wainwright, 2019, Thm. 2.26): the bound is the same for two arguments and for a thousand. It constrains the tails of $f({\mathbf z})$ and does not state that $f({\mathbf z})$ has a normal distribution. The Euclidean norm is an example with $L = 1$: the length $\Vert {{\mathbf z}} \Vert$ has expectation at most $\sqrt{d}$ and deviates from it by more than $t$ with probability at most $2\exp(-t^{2}/2)$, however large $d$ is.

Among all random vectors with a given covariance matrix ${\mathbf C}$, the Gaussian random vector ${\mathbf x} \sim \mathcal{N}\left({\boldsymbol \mu},{\mathbf C}\right)$ maximizes the differential entropy (Cover and Thomas, 2006, Thm. 8.6.5). This property is one reason why additive Gaussian measurement noise is a widely used probabilistic model in the statistical analysis of machine learning methods: when only the variance of the noise is known, a Gaussian is the least restrictive assumption.

**See also:** multivariate normal distribution, Gaussian process, probabilistic model, central limit theorem, differential entropy, concentration inequality, sub-Gaussian random variable.

## References

- Bertsekas and Tsitsiklis (2008). Introduction to Probability. *Athena Scientific*.
- Gray (2009). Probability, Random Processes, and Ergodic Properties. *Springer Science+Business Media*. <https://doi.org/10.1007/978-1-4419-1090-5>
- Papoulis and Pillai (2002). Probability, Random Variables, and Stochastic Processes. *McGraw-Hill Higher Education*.
- Ross (2014). A First Course in Probability. *Pearson Education*.
- Lapidoth (2017). A Foundation in Digital Communication. *Cambridge Univ. Press*.
- Rasmussen and Williams (2006). Gaussian Processes for Machine Learning. *MIT Press*. <https://doi.org/10.7551/mitpress/3206.001.0001>
- Wainwright (2019). High-Dimensional Statistics: A Non-Asymptotic Viewpoint. *Cambridge Univ. Press*.
- Cover and Thomas (2006). Elements of Information Theory. *Wiley*. <https://doi.org/10.1002/047174882X>

## Cite this entry

```bibtex
@misc{dictml_gaussrv,
  author = {Jung, Alexander},
  editor = {Olioumtsevits, Konstantina and Schnoor, Ekkehard},
  title = {Gaussian random variable (Gaussian RV)},
  howpublished = {Dictionary of Applied Machine Learning (course edition)},
  year = {2026},
  doi = {10.5281/zenodo.21569296},
  note = {ISBN 978-952-64-3013-3, CC BY 4.0, retrieved 2026-11-04},
  url = {https://dictionaryofml.org/terms/gaussrv.html}
}
```