AP Statistics Curriculum 2007 MultivariateNormal

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EBook - Multivariate Normal Distribution

The multivariate normal distribution, or multivariate Gaussian distribution, is a generalization of the univariate (one-dimensional) normal distribution to higher dimensions. A random vector is said to be multivariate normally distributed if every linear combination of its components has a univariate normal distribution. The multivariate normal distribution may be used to study different associations (e.g., correlations) between real-valued random variables.


In k-dimensions, a random vector X = (X_1, \cdots, X_k) is multivariate normally distributed if it satisfies any one of the following equivalent conditions (Gut, 2009):

  • Every linear combination of its components Y = a1X1 + … + akXk is normally distributed. In other words, for any constant vector a\in R^k, the linear combination (which is univariate random variable) Y = a^TX = \sum_{i=1}^{k}{a_iX_i} has a univariate normal distribution.
  • There exists a random -vector Z, whose components are independent normal random variables, a k-vector μ, and a k×ℓ matrix A, such that X = AZ + μ. Here is the rank of the variance-covariance matrix.
  • There is a k-vector μ and a symmetric, nonnegative-definite k×k matrix Σ, such that the characteristic function of X is

    \varphi_X(u) = \exp\Big( iu^T\mu - \tfrac{1}{2} u^T\Sigma u \Big).
  • When the support of X is the entire space Rk, there exists a k-vector μ and a symmetric positive-definite k×k variance-covariance matrix Σ, such that the probability density function of X can be expressed as

    f_X(x) = \frac{1}{ (2\pi)^{k/2}|\Sigma|^{1/2} }
             \exp\!\Big( {-\tfrac{1}{2}}(x-\mu)'\Sigma^{-1}(x-\mu) \Big)
  , where |Σ| is the determinant of Σ, and where (2π)k/2|Σ|1/2 = |2πΣ|1/2. This formulation reduces to the density of the univariate normal distribution if Σ is a scalar (i.e., a 1×1 matrix).

If the variance-covariance matrix is singular, the corresponding distribution has no density. An example of this case is the distribution of the vector of residual-errors in the ordinary least squares regression. Note also that the Xi are in general not independent; they can be seen as the result of applying the matrix A to a collection of independent Gaussian variables Z.

Bivariate (2D) case

See the SOCR Bivariate Normal Distribution Activity and corresponding Webapp.

In 2-dimensions, the nonsingular bi-variate Normal distribution with (k = rank(Σ) = 2), the probability density function of a (bivariate) vector (X,Y) is

    f(x,y) =
      \frac{1}{2 \pi  \sigma_x \sigma_y \sqrt{1-\rho^2}}
          \frac{(x-\mu_x)^2}{\sigma_x^2} +
          \frac{(y-\mu_y)^2}{\sigma_y^2} -
          \frac{2\rho(x-\mu_x)(y-\mu_y)}{\sigma_x \sigma_y}

where ρ is the correlation between X and Y. In this case,

    \mu = \begin{pmatrix} \mu_x \\ \mu_y \end{pmatrix}, \quad
    \Sigma = \begin{pmatrix} \sigma_x^2 & \rho \sigma_x \sigma_y \\
                             \rho \sigma_x \sigma_y  & \sigma_y^2 \end{pmatrix}.

In the bivariate case, the first equivalent condition for multivariate normality is less restrictive: it is sufficient to verify that countably many distinct linear combinations of X and Y are normal in order to conclude that the vector [X,Y]T is bivariate normal.


Normally distributed and independent

If X and Y are normally distributed and independent, this implies they are "jointly normally distributed", hence, the pair (XY) must have bivariate normal distribution. However, a pair of jointly normally distributed variables need not be independent - they could be correlated.

Two normally distributed random variables need not be jointly bivariate normal

The fact that two random variables X and Y both have a normal distribution does not imply that the pair (XY) has a joint normal distribution. A simple example is provided below:

Let X ~ N(0,1).
Let Y = \begin{cases} X,& |X| > 1.33,\\
-X,& |X| \leq 1.33.\end{cases}

Then, both X and Y are individually Normally distributed; however, the pair (X,Y) is not jointly bivariate Normal distributed (of course, the constant c=1.33 is not special, any other non-trivial constant also works).

Furthermore, as X and Y are not independent, the sum Z = X+Y is not guaranteed to be a (univariate) Normal variable. In this case, it's clear that Z is not Normal:

Z = \begin{cases} 0,& |X| \leq 1.33,\\
2X,& |X| > 1.33.\end{cases}


This SOCR activity demonstrates the use of 2D Gaussian distribution, expectation maximization and mixture modeling for classification of points (objects) in 2D.



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