**Univariate case**

A random variable *x* has normal distribution if its probability density function (pdf) can be expressed as

Here *e* is the constant 2.7183…, and *π* is the constant 3.1415…

The normal distribution is completely determined by the parameters *μ* (mean) and *σ* (standard deviation). We use the abbreviation *N*(μ, *σ*) to refer to a normal distribution with mean μ and standard deviation *σ, *although for comparison with the multivariate case it would actually be better to use the abbreviation *N*(μ, *σ*^{2}) where *σ*^{2} is the variance.

**Multivariate case**

**Definition 1**: A random vector *X* has a **multivariate normal distribution** with vector mean *μ* and covariance matrix *Σ*, written *X* ~ *N*(*μ*, *Σ*) if *X* has the following joint probability density function:

Here |*Σ*| is the determinant of the population covariance matrix Σ. The exponent of *e* consists of the product of the transpose of *X* – *μ*, the inverse of *Σ* and *X* – *μ*, which has dimension (1 × *k*) × (*k* × *k*) × (*k* × 1) = 1 × 1, i.e. a scalar. Thus *f*(*X*) yields a single value. The coefficient (2π)^{k} |Σ| can also be expressed as |2*πΣ|.*

**Definition 2**: The expression

which appears in the exponent of *e,* is called the squared **Mahalanobis distance** between *X* and *μ*. We can also define the squared Mahalanobis distance for a sample to be

Where *S* is the sample covariance matrix and *X̄* is the sample mean vector. In MANOVA we give an example of how to calculate this value and also introduce the supplemental function **MDistSq** which calculates this value automatically.

**Observation**: If *k* = 1 then the above definition is equivalent to the univariate normal distribution. If *k* = 2 the result is a three dimensional bell shaped curve (as described in Figure 1).

**Property 1**: If *X* ~ *N*(*μ*, *Σ*) where all the *x _{j}* in

*X*are independent, then the population covariance matrix is a diagonal matrix [

*a*] with

_{ij}*a*= for all

_{jj}*j*and

*a*= 0 for all

_{ij}*i*≠

*j*, and so the joint probability function simplifies to

where each is the univariate normal pdf of *x _{j}* with mean

*μ*and standard deviation

_{j}*σ*.

_{j}**Property 2**: If y = = *C*^{T}*X*, where *C* = the *k* × 1 vector [*c _{j}*], and

*X*~

*N*(

*μ, Σ*) then y has normal distribution with mean =

*C*

^{T}μ

*and variance =*

*C*

^{T}

*ΣC*; i.e. y ~

*N*(

*C*).

^{T}μ,C^{T}ΣC**Observation**: The unbiased estimates for population mean and population variance are given by the sample mean = *C*^{T}*X̄* and sample variance = *C*^{T}*SC*, where *s _{ij} *= cov(

*x*).

_{i}, x_{j}**Observation**: When *k* = 2, the joint pdf of *X* depends on the parameters *μ*_{1}, *μ*_{2}, *σ*_{1}, *σ*_{2}, and *ρ*. A plot of the distribution for different values of the correlation coefficient *ρ* is displayed in Figure 1.

**Figure 1 – ****Bivariate normal density function**

**Observation**: Suppose *X* has a multivariate normal distribution. For any constant *c*, the set of points *X* which have a Mahalanobis distance from *μ* of *c* sketches out a *k*-dimensional ellipse. The value of the probability density function at all these points is the constant

Let’s take a look at the situation where *k* = 2. In this case, we have

Finally, note that the equation

is an ellipse with foci at μ = (μ_{1}, μ_{2}).

**Observation**: Note that when *ρ* = 0, indicating that *x*_{1} and *x*_{2} are uncorrelated, then the ellipse takes the form (*z*_{1} – *z*_{2})^{2} = *c*^{2} which is a circle. When *ρ* = ±1, indicating that *x*_{1} and *x*_{2} are completely correlated, then the ellipse becomes a straight line.

**Observation**: Using the above calculations, when *k* = 2 the multivariate normal pdf is

**Property 3**: If *X* ~ *N*(*μ, Σ*), then the squared Mahalanobis distance between *X* and *μ* has a chi-square distribution with *k* degrees of freedom.

**Observation**: This property is an extension of Corollary 1 of Chi-square Distribution. We can interpret the property as follows. Let *c* = the critical value of the chi-square distribution with *k* degrees of freedom for α = .05. Then the probability that *X* will fall within the ellipse defined by *c*, i.e. (*X–μ*)^{T} *Σ*^{-1} (*X–μ*) = *c*^{2}, is 1 – *α* = .95.

**NOTE**: The Real Statistics supplemental functions and data analysis tools are not yet available. They will be provided in the Real Statistics Resource Pack Release 2.0 which will be available shortly.