Normal and Multivariate Normal Distributions

Gaussian models for continuous variables and correlated vectors

Statistics

Probability

Distributions

Multivariate Analysis

Univariate and multivariate normal distributions, linear transformations, conditional laws, Mahalanobis distance, assumptions, and biological interpretation.

Author

Jiahong Sun

Published

September 12, 2026

Modified

September 12, 2026

1 Topic at a Glance

One-sentence summary: A normal distribution models a continuous quantity with symmetric Gaussian variation, while a multivariate normal distribution describes several such quantities through a mean vector and covariance matrix.

Item	Univariate normal	Multivariate normal
Random quantity	Scalar $X$	Vector $\mathbf X\in\mathbb R^p$
Parameters	Mean $\mu$ and variance $\sigma^2>0$	Mean $\boldsymbol\mu$ and covariance $\boldsymbol\Sigma\succeq0$
Support	$\mathbb R$	Usually $\mathbb R^p$; a lower-dimensional affine subspace if singular
Shape	Symmetric bell curve	Elliptical contours when $\boldsymbol\Sigma$ is positive definite
Typical applications	Measurement errors, transformed expression summaries, sampling approximations	Correlated traits, assay panels, latent Gaussian models
Prerequisites	Expectation, variance, and moments	Covariance and correlation; transformations of random variables

2 The Univariate Normal Distribution

A continuous random variable has a normal distribution with mean $\mu\in\mathbb R$ and variance $\sigma^2>0$, written

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

when its density is

\[ \boxed{ f_X(x)=\frac{1}{\sigma\sqrt{2\pi}} \exp\left\{-\frac{(x-\mu)^2}{2\sigma^2}\right\}, \qquad -\infty<x<\infty. } \]

The density is symmetric about $\mu$, so its mean, median, and mode coincide. Its moments include

\[ E(X)=\mu, \qquad \operatorname{Var}(X)=\sigma^2, \qquad E\{(X-\mu)^3\}=0, \qquad E\{(X-\mu)^4\}=3\sigma^4. \]

Thus normal excess kurtosis is zero. Symmetry alone does not imply normality: many symmetric distributions have different centers, tails, or kurtosis.

2.1 Standardization and probabilities

The affine transformation

\[ Z=\frac{X-\mu}{\sigma} \]

has the standard normal distribution $N(0,1)$. If $\Phi(z)=P(Z\leq z)$, then

\[ P(a<X\leq b) =\Phi\left(\frac{b-\mu}{\sigma}\right) -\Phi\left(\frac{a-\mu}{\sigma}\right). \]

There is no elementary antiderivative for the Gaussian density, so normal probabilities and quantiles are evaluated numerically. The familiar approximate intervals $\mu\pm\sigma$, $\mu\pm2\sigma$, and $\mu\pm3\sigma$ contain about 68%, 95%, and 99.7% of the distribution, respectively; these are properties of the model, not universal empirical rules.

3 Why Gaussian Models Recur

Normal variables are closed under affine transformations. If $X\sim N(\mu,\sigma^2)$, then

\[ a+bX\sim N(a+b\mu,b^2\sigma^2). \]

If $X_1,\ldots,X_n$ are independent normals, every linear combination is normal:

\[ \sum_{i=1}^n a_iX_i \sim N\left(\sum_i a_i\mu_i,\sum_i a_i^2\sigma_i^2\right). \]

Gaussian laws also arise as limits for standardized sums under central-limit-theorem conditions. That fact can justify an approximation to a statistic even when individual observations are not normal. It does not imply that every biological measurement is Gaussian, that small samples are protected from skewness or outliers, or that dependence can be ignored.

The moment-generating function makes the closure property transparent:

\[ M_X(t)=E(e^{tX}) =\exp\left(\mu t+\frac12\sigma^2t^2\right). \]

Products of moment-generating functions for independent sums add their means and variances in the exponent.

4 The Multivariate Normal Distribution

A random vector $\mathbf X\in\mathbb R^p$ is multivariate normal if every linear combination $\mathbf a^\mathsf T\mathbf X$ is univariate normal. We write

\[ \mathbf X\sim N_p(\boldsymbol\mu,\boldsymbol\Sigma), \]

where

\[ E(\mathbf X)=\boldsymbol\mu, \qquad \operatorname{Cov}(\mathbf X)=\boldsymbol\Sigma. \]

The covariance matrix must be symmetric and positive semidefinite. If it is positive definite, the joint density is

\[ \boxed{ f_{\mathbf X}(\mathbf x) =\frac{1}{(2\pi)^{p/2}|\boldsymbol\Sigma|^{1/2}} \exp\left\{-\frac12 (\mathbf x-\boldsymbol\mu)^\mathsf T \boldsymbol\Sigma^{-1} (\mathbf x-\boldsymbol\mu)\right\}. } \]

Equal-density contours are ellipsoids centered at $\boldsymbol\mu$. Their orientation and axis lengths are determined by the eigenvectors and eigenvalues of $\boldsymbol\Sigma$.

If $\boldsymbol\Sigma$ is singular, the linear-combination definition still gives a valid multivariate normal distribution, but no density exists with respect to $p$-dimensional volume. The vector lies on a lower-dimensional affine subspace. This occurs, for example, when one component is an exact linear combination of others.

4.1 Constructing a Gaussian vector

Let $\mathbf Z\sim N_r(\mathbf0,\mathbf I_r)$ and define

\[ \mathbf X=\boldsymbol\mu+\mathbf A\mathbf Z. \]

Then

\[ \mathbf X\sim N_p(\boldsymbol\mu,\mathbf A\mathbf A^\mathsf T). \]

Every multivariate normal vector can be represented this way. A Cholesky factor is convenient when $\boldsymbol\Sigma$ is positive definite; an eigenvalue factorization also handles positive-semidefinite matrices.

5 Marginals, Linear Maps, and Conditionals

Any subvector of a multivariate normal vector is multivariate normal. More generally, for a fixed matrix $\mathbf B$ and vector $\mathbf c$,

\[ \boxed{ \mathbf B\mathbf X+\mathbf c \sim N\left(\mathbf B\boldsymbol\mu+\mathbf c, \mathbf B\boldsymbol\Sigma\mathbf B^\mathsf T\right). } \]

Partition the vector and its parameters as

\[ \mathbf X= \begin{pmatrix}\mathbf X_1\\\mathbf X_2\end{pmatrix}, \quad \boldsymbol\mu= \begin{pmatrix}\boldsymbol\mu_1\\\boldsymbol\mu_2\end{pmatrix}, \quad \boldsymbol\Sigma= \begin{pmatrix} \boldsymbol\Sigma_{11}&\boldsymbol\Sigma_{12}\\ \boldsymbol\Sigma_{21}&\boldsymbol\Sigma_{22} \end{pmatrix}. \]

When $\boldsymbol\Sigma_{22}$ is invertible,

\[ \mathbf X_1\mid\mathbf X_2=\mathbf x_2 \sim N\left( \boldsymbol\mu_1+ \boldsymbol\Sigma_{12}\boldsymbol\Sigma_{22}^{-1} (\mathbf x_2-\boldsymbol\mu_2), \boldsymbol\Sigma_{11}- \boldsymbol\Sigma_{12}\boldsymbol\Sigma_{22}^{-1} \boldsymbol\Sigma_{21} \right). \]

Conditioning shifts the mean linearly and reduces, or leaves unchanged, the covariance. The conditional covariance does not depend on the observed value $\mathbf x_2$.

Within a jointly multivariate normal vector, zero cross-covariance implies independence:

\[ \boldsymbol\Sigma_{12}=\mathbf0 \quad\Longleftrightarrow\quad \mathbf X_1\perp\!\!\!\perp\mathbf X_2. \]

This equivalence is special to the joint Gaussian family. Variables can each have normal marginal distributions and still fail to be jointly normal; in that case zero covariance need not imply independence.

6 Mahalanobis Distance and a Chi-squared Connection

For positive-definite $\boldsymbol\Sigma$, whiten the vector as

\[ \mathbf Z=\boldsymbol\Sigma^{-1/2}(\mathbf X-\boldsymbol\mu). \]

Then $\mathbf Z\sim N_p(\mathbf0,\mathbf I)$, and the squared Mahalanobis distance is

\[ \boxed{ D^2=(\mathbf X-\boldsymbol\mu)^\mathsf T \boldsymbol\Sigma^{-1}(\mathbf X-\boldsymbol\mu) =\mathbf Z^\mathsf T\mathbf Z \sim\chi_p^2. } \]

This result converts an elliptical distance into a probability scale and links multivariate normal geometry to chi-squared, Student’s t, and F distributions. Estimating $\boldsymbol\mu$ and $\boldsymbol\Sigma$ from the same data changes the exact reference distribution; treating estimated parameters as known can make an outlier rule too optimistic.

7 Worked Example: Two Correlated Assay Summaries

Suppose two standardized log-expression summaries from one biological specimen are idealized as

\[ \begin{pmatrix}X_1\\X_2\end{pmatrix} \sim N_2\left[ \begin{pmatrix}0\\0\end{pmatrix}, \begin{pmatrix}1&0.8\\0.8&1\end{pmatrix} \right]. \]

The difference $D=X_1-X_2$ is normal because it is a linear combination, with

\[ E(D)=0, \qquad \operatorname{Var}(D)=1+1-2(0.8)=0.4. \]

Ignoring the covariance would give variance 2, five times too large. Conversely, the conditional distribution of $X_1$ given $X_2=x_2$ is

\[ X_1\mid X_2=x_2\sim N(0.8x_2,1-0.8^2)=N(0.8x_2,0.36). \]

The calculation describes within-specimen joint measurement variation. It does not turn two assay variables, many genes, or repeated technical measurements into independent biological replicates. Population claims still require independently sampled biological units and a design that represents the target population.

8 Modeling Checks and Failure Modes

Assumption or feature	Why it matters	What to inspect	Possible response
Symmetric, light-tailed marginals	Gaussian tails assign very low probability to extremes	Histograms, quantile plots, robust summaries	Transform, use a heavy-tailed model, or report robust analysis
Elliptical joint shape	Mean and covariance determine Gaussian dependence	Pair plots, projections, residual structure	Mixture, copula, nonlinear, or latent-state model
Stable covariance	Conditional laws and distances depend on $\boldsymbol\Sigma$	Covariance across groups, batches, and mean levels	Model group-specific or mean-dependent covariance
Independent experimental units	Standard likelihood factorization requires the sampling structure	Pedigrees, repeated measures, batches, nested observations	Mixed, clustered, longitudinal, or hierarchical model
Adequate dimension-to-sample ratio	Sample covariance can be unstable or singular	Eigenvalues and sensitivity to observations	Regularization, dimension reduction, or more biological samples

Normal-looking one-dimensional projections do not prove joint normality. A mixture of Gaussian subpopulations can be multimodal or heavy-tailed, and batch effects can create correlations absent within each batch. With $p$ near or above the number of independent samples, the empirical covariance matrix is poorly determined even if the Gaussian model itself is correct.

9 Common Pitfalls

Calling any bell-shaped histogram normal: the tail behavior and data-generating mechanism also matter.
Using standard deviation as a universal 95% rule: the familiar coverage percentages are Gaussian model properties.
Equating marginal normality with joint normality: Gaussian marginals do not determine the dependence structure.
Equating zero correlation with independence outside a joint Gaussian model: the implication is distribution-specific.
Ignoring singular or unstable covariance: inversion and Mahalanobis distances require more than a visually plausible matrix.
Counting variables or technical observations as replicates: replication is defined by independently sampled experimental units.

10 Connections to the Knowledge Map

Random variables and probability distributions defines densities, CDFs, and quantiles.
Expectation, variance, and moments supplies moment and moment-generating-function calculations.
Joint, marginal, and conditional distributions provides the operations used for subvectors and conditioning.
Covariance and correlation develops covariance matrices and their interpretation.
Transformations of random variables explains affine maps and Jacobian-based derivations.
Chi-squared, Student’s t, and F distributions develops quadratic forms and exact normal-sample pivots.

11 Take-home Messages

A univariate normal law is determined by its mean and variance and becomes standard normal after centering and scaling.
A multivariate normal law is characterized by normality of every linear combination and by a mean vector and covariance matrix.
Marginal, conditional, and affine-transformed distributions remain Gaussian.
Zero covariance implies independence only under a jointly Gaussian model, not merely normal marginals.
Mahalanobis distance has a chi-squared law when the mean and covariance are known and the model is multivariate normal.
Outliers, mixtures, nonlinear dependence, batch effects, and limited biological replication can invalidate simple Gaussian analyses.

12 References

Anderson, T. W. An Introduction to Multivariate Statistical Analysis, 3rd ed. Wiley, 2003.
Mardia, K. V., Kent, J. T., and Bibby, J. M. Multivariate Analysis. Academic Press, 1979.
Tong, Y. L. The Multivariate Normal Distribution. Springer, 1990. https://doi.org/10.1007/978-1-4613-9655-0.
NIST/SEMATECH. “What Do We Mean by ‘Normal’ Data?” e-Handbook of Statistical Methods. https://www.itl.nist.gov/div898/handbook/pmc/section5/pmc51.htm.
NIST/SEMATECH. “The Multivariate Normal Distribution.” e-Handbook of Statistical Methods. https://www.itl.nist.gov/div898/handbook/pmc/section5/pmc542.htm.
R Core Team. “The Normal Distribution.” R Statistical Functions Documentation. https://stat.ethz.ch/R-manual/R-devel/library/stats/html/Normal.html.

--- title: "Normal and Multivariate Normal Distributions" subtitle: "Gaussian models for continuous variables and correlated vectors" description: "Univariate and multivariate normal distributions, linear transformations, conditional laws, Mahalanobis distance, assumptions, and biological interpretation." author: "Jiahong Sun" date: "2026-09-12" date-modified: "2026-09-12" categories: [Statistics, Probability, Distributions, Multivariate Analysis] toc: true number-sections: true draft: false --- # Topic at a Glance > **One-sentence summary:** A normal distribution models a continuous quantity with symmetric Gaussian variation, while a multivariate normal distribution describes several such quantities through a mean vector and covariance matrix. | Item | Univariate normal | Multivariate normal | |---|---|---| | Random quantity | Scalar $X$ | Vector $\mathbf X\in\mathbb R^p$ | | Parameters | Mean $\mu$ and variance $\sigma^2>0$ | Mean $\boldsymbol\mu$ and covariance $\boldsymbol\Sigma\succeq0$ | | Support | $\mathbb R$ | Usually $\mathbb R^p$; a lower-dimensional affine subspace if singular | | Shape | Symmetric bell curve | Elliptical contours when $\boldsymbol\Sigma$ is positive definite | | Typical applications | Measurement errors, transformed expression summaries, sampling approximations | Correlated traits, assay panels, latent Gaussian models | | Prerequisites | [Expectation, variance, and moments](expectation-variance-and-moments.qmd) | [Covariance and correlation](covariance-and-correlation.qmd); [transformations of random variables](transformations-of-random-variables.qmd) | # The Univariate Normal Distribution A continuous random variable has a normal distribution with mean $\mu\in\mathbb R$ and variance $\sigma^2>0$, written $$ X\sim N(\mu,\sigma^2), $$ when its density is $$ \boxed{ f_X(x)=\frac{1}{\sigma\sqrt{2\pi}} \exp\left\{-\frac{(x-\mu)^2}{2\sigma^2}\right\}, \qquad -\infty<x<\infty. } $$ The density is symmetric about $\mu$, so its mean, median, and mode coincide. Its moments include $$ E(X)=\mu, \qquad \operatorname{Var}(X)=\sigma^2, \qquad E\{(X-\mu)^3\}=0, \qquad E\{(X-\mu)^4\}=3\sigma^4. $$ Thus normal excess kurtosis is zero. Symmetry alone does not imply normality: many symmetric distributions have different centers, tails, or kurtosis. ## Standardization and probabilities The affine transformation $$ Z=\frac{X-\mu}{\sigma} $$ has the standard normal distribution $N(0,1)$. If $\Phi(z)=P(Z\leq z)$, then $$ P(a<X\leq b) =\Phi\left(\frac{b-\mu}{\sigma}\right) -\Phi\left(\frac{a-\mu}{\sigma}\right). $$ There is no elementary antiderivative for the Gaussian density, so normal probabilities and quantiles are evaluated numerically. The familiar approximate intervals $\mu\pm\sigma$, $\mu\pm2\sigma$, and $\mu\pm3\sigma$ contain about 68%, 95%, and 99.7% of the distribution, respectively; these are properties of the model, not universal empirical rules. # Why Gaussian Models Recur Normal variables are closed under affine transformations. If $X\sim N(\mu,\sigma^2)$, then $$ a+bX\sim N(a+b\mu,b^2\sigma^2). $$ If $X_1,\ldots,X_n$ are independent normals, every linear combination is normal: $$ \sum_{i=1}^n a_iX_i \sim N\left(\sum_i a_i\mu_i,\sum_i a_i^2\sigma_i^2\right). $$ Gaussian laws also arise as limits for standardized sums under central-limit-theorem conditions. That fact can justify an approximation to a statistic even when individual observations are not normal. It does **not** imply that every biological measurement is Gaussian, that small samples are protected from skewness or outliers, or that dependence can be ignored. The moment-generating function makes the closure property transparent: $$ M_X(t)=E(e^{tX}) =\exp\left(\mu t+\frac12\sigma^2t^2\right). $$ Products of moment-generating functions for independent sums add their means and variances in the exponent. # The Multivariate Normal Distribution A random vector $\mathbf X\in\mathbb R^p$ is multivariate normal if every linear combination $\mathbf a^\mathsf T\mathbf X$ is univariate normal. We write $$ \mathbf X\sim N_p(\boldsymbol\mu,\boldsymbol\Sigma), $$ where $$ E(\mathbf X)=\boldsymbol\mu, \qquad \operatorname{Cov}(\mathbf X)=\boldsymbol\Sigma. $$ The covariance matrix must be symmetric and positive semidefinite. If it is positive definite, the joint density is $$ \boxed{ f_{\mathbf X}(\mathbf x) =\frac{1}{(2\pi)^{p/2}|\boldsymbol\Sigma|^{1/2}} \exp\left\{-\frac12 (\mathbf x-\boldsymbol\mu)^\mathsf T \boldsymbol\Sigma^{-1} (\mathbf x-\boldsymbol\mu)\right\}. } $$ Equal-density contours are ellipsoids centered at $\boldsymbol\mu$. Their orientation and axis lengths are determined by the eigenvectors and eigenvalues of $\boldsymbol\Sigma$. If $\boldsymbol\Sigma$ is singular, the linear-combination definition still gives a valid multivariate normal distribution, but no density exists with respect to $p$-dimensional volume. The vector lies on a lower-dimensional affine subspace. This occurs, for example, when one component is an exact linear combination of others. ## Constructing a Gaussian vector Let $\mathbf Z\sim N_r(\mathbf0,\mathbf I_r)$ and define $$ \mathbf X=\boldsymbol\mu+\mathbf A\mathbf Z. $$ Then $$ \mathbf X\sim N_p(\boldsymbol\mu,\mathbf A\mathbf A^\mathsf T). $$ Every multivariate normal vector can be represented this way. A Cholesky factor is convenient when $\boldsymbol\Sigma$ is positive definite; an eigenvalue factorization also handles positive-semidefinite matrices. # Marginals, Linear Maps, and Conditionals Any subvector of a multivariate normal vector is multivariate normal. More generally, for a fixed matrix $\mathbf B$ and vector $\mathbf c$, $$ \boxed{ \mathbf B\mathbf X+\mathbf c \sim N\left(\mathbf B\boldsymbol\mu+\mathbf c, \mathbf B\boldsymbol\Sigma\mathbf B^\mathsf T\right). } $$ Partition the vector and its parameters as $$ \mathbf X= \begin{pmatrix}\mathbf X_1\\\mathbf X_2\end{pmatrix}, \quad \boldsymbol\mu= \begin{pmatrix}\boldsymbol\mu_1\\\boldsymbol\mu_2\end{pmatrix}, \quad \boldsymbol\Sigma= \begin{pmatrix} \boldsymbol\Sigma_{11}&\boldsymbol\Sigma_{12}\\ \boldsymbol\Sigma_{21}&\boldsymbol\Sigma_{22} \end{pmatrix}. $$ When $\boldsymbol\Sigma_{22}$ is invertible, $$ \mathbf X_1\mid\mathbf X_2=\mathbf x_2 \sim N\left( \boldsymbol\mu_1+ \boldsymbol\Sigma_{12}\boldsymbol\Sigma_{22}^{-1} (\mathbf x_2-\boldsymbol\mu_2), \boldsymbol\Sigma_{11}- \boldsymbol\Sigma_{12}\boldsymbol\Sigma_{22}^{-1} \boldsymbol\Sigma_{21} \right). $$ Conditioning shifts the mean linearly and reduces, or leaves unchanged, the covariance. The conditional covariance does not depend on the observed value $\mathbf x_2$. Within a jointly multivariate normal vector, zero cross-covariance implies independence: $$ \boldsymbol\Sigma_{12}=\mathbf0 \quad\Longleftrightarrow\quad \mathbf X_1\perp\!\!\!\perp\mathbf X_2. $$ This equivalence is special to the joint Gaussian family. Variables can each have normal marginal distributions and still fail to be jointly normal; in that case zero covariance need not imply independence. # Mahalanobis Distance and a Chi-squared Connection For positive-definite $\boldsymbol\Sigma$, whiten the vector as $$ \mathbf Z=\boldsymbol\Sigma^{-1/2}(\mathbf X-\boldsymbol\mu). $$ Then $\mathbf Z\sim N_p(\mathbf0,\mathbf I)$, and the squared Mahalanobis distance is $$ \boxed{ D^2=(\mathbf X-\boldsymbol\mu)^\mathsf T \boldsymbol\Sigma^{-1}(\mathbf X-\boldsymbol\mu) =\mathbf Z^\mathsf T\mathbf Z \sim\chi_p^2. } $$ This result converts an elliptical distance into a probability scale and links multivariate normal geometry to [chi-squared, Student's t, and F distributions](chi-squared-students-t-and-f-distributions.qmd). Estimating $\boldsymbol\mu$ and $\boldsymbol\Sigma$ from the same data changes the exact reference distribution; treating estimated parameters as known can make an outlier rule too optimistic. # Worked Example: Two Correlated Assay Summaries Suppose two standardized log-expression summaries from one biological specimen are idealized as $$ \begin{pmatrix}X_1\\X_2\end{pmatrix} \sim N_2\left[ \begin{pmatrix}0\\0\end{pmatrix}, \begin{pmatrix}1&0.8\\0.8&1\end{pmatrix} \right]. $$ The difference $D=X_1-X_2$ is normal because it is a linear combination, with $$ E(D)=0, \qquad \operatorname{Var}(D)=1+1-2(0.8)=0.4. $$ Ignoring the covariance would give variance 2, five times too large. Conversely, the conditional distribution of $X_1$ given $X_2=x_2$ is $$ X_1\mid X_2=x_2\sim N(0.8x_2,1-0.8^2)=N(0.8x_2,0.36). $$ The calculation describes within-specimen joint measurement variation. It does not turn two assay variables, many genes, or repeated technical measurements into independent biological replicates. Population claims still require independently sampled biological units and a design that represents the target population. # Modeling Checks and Failure Modes | Assumption or feature | Why it matters | What to inspect | Possible response | |---|---|---|---| | Symmetric, light-tailed marginals | Gaussian tails assign very low probability to extremes | Histograms, quantile plots, robust summaries | Transform, use a heavy-tailed model, or report robust analysis | | Elliptical joint shape | Mean and covariance determine Gaussian dependence | Pair plots, projections, residual structure | Mixture, copula, nonlinear, or latent-state model | | Stable covariance | Conditional laws and distances depend on $\boldsymbol\Sigma$ | Covariance across groups, batches, and mean levels | Model group-specific or mean-dependent covariance | | Independent experimental units | Standard likelihood factorization requires the sampling structure | Pedigrees, repeated measures, batches, nested observations | Mixed, clustered, longitudinal, or hierarchical model | | Adequate dimension-to-sample ratio | Sample covariance can be unstable or singular | Eigenvalues and sensitivity to observations | Regularization, dimension reduction, or more biological samples | Normal-looking one-dimensional projections do not prove joint normality. A mixture of Gaussian subpopulations can be multimodal or heavy-tailed, and batch effects can create correlations absent within each batch. With $p$ near or above the number of independent samples, the empirical covariance matrix is poorly determined even if the Gaussian model itself is correct. # Common Pitfalls 1. **Calling any bell-shaped histogram normal:** the tail behavior and data-generating mechanism also matter. 2. **Using standard deviation as a universal 95% rule:** the familiar coverage percentages are Gaussian model properties. 3. **Equating marginal normality with joint normality:** Gaussian marginals do not determine the dependence structure. 4. **Equating zero correlation with independence outside a joint Gaussian model:** the implication is distribution-specific. 5. **Ignoring singular or unstable covariance:** inversion and Mahalanobis distances require more than a visually plausible matrix. 6. **Counting variables or technical observations as replicates:** replication is defined by independently sampled experimental units. # Connections to the Knowledge Map - [Random variables and probability distributions](random-variables-and-probability-distributions.qmd) defines densities, CDFs, and quantiles. - [Expectation, variance, and moments](expectation-variance-and-moments.qmd) supplies moment and moment-generating-function calculations. - [Joint, marginal, and conditional distributions](joint-marginal-and-conditional-distributions.qmd) provides the operations used for subvectors and conditioning. - [Covariance and correlation](covariance-and-correlation.qmd) develops covariance matrices and their interpretation. - [Transformations of random variables](transformations-of-random-variables.qmd) explains affine maps and Jacobian-based derivations. - [Chi-squared, Student's t, and F distributions](chi-squared-students-t-and-f-distributions.qmd) develops quadratic forms and exact normal-sample pivots. # Take-home Messages 1. A univariate normal law is determined by its mean and variance and becomes standard normal after centering and scaling. 2. A multivariate normal law is characterized by normality of every linear combination and by a mean vector and covariance matrix. 3. Marginal, conditional, and affine-transformed distributions remain Gaussian. 4. Zero covariance implies independence only under a jointly Gaussian model, not merely normal marginals. 5. Mahalanobis distance has a chi-squared law when the mean and covariance are known and the model is multivariate normal. 6. Outliers, mixtures, nonlinear dependence, batch effects, and limited biological replication can invalidate simple Gaussian analyses. # References - Anderson, T. W. *An Introduction to Multivariate Statistical Analysis*, 3rd ed. Wiley, 2003. - Mardia, K. V., Kent, J. T., and Bibby, J. M. *Multivariate Analysis*. Academic Press, 1979. - Tong, Y. L. *The Multivariate Normal Distribution*. Springer, 1990. [https://doi.org/10.1007/978-1-4613-9655-0](https://doi.org/10.1007/978-1-4613-9655-0). - NIST/SEMATECH. “What Do We Mean by ‘Normal’ Data?” *e-Handbook of Statistical Methods*. [https://www.itl.nist.gov/div898/handbook/pmc/section5/pmc51.htm](https://www.itl.nist.gov/div898/handbook/pmc/section5/pmc51.htm). - NIST/SEMATECH. “The Multivariate Normal Distribution.” *e-Handbook of Statistical Methods*. [https://www.itl.nist.gov/div898/handbook/pmc/section5/pmc542.htm](https://www.itl.nist.gov/div898/handbook/pmc/section5/pmc542.htm). - R Core Team. “The Normal Distribution.” *R Statistical Functions Documentation*. [https://stat.ethz.ch/R-manual/R-devel/library/stats/html/Normal.html](https://stat.ethz.ch/R-manual/R-devel/library/stats/html/Normal.html).