Multivariate normal distribution

1. Sablon:Matematika többváltozós normális eloszlás

The multivariate normal distribution is a generalization of the one-dimensional normal distribution to higher dimensions. It is defined for a vector of random variables and is characterized by its mean vector and covariance matrix.

Definition

A random vector $𝐗=(X_1,X_2,\dots ,X_k{)}^T$ follows a multivariate normal distribution if every linear combination of its components has a univariate normal distribution. The notation for a multivariate normal distribution is:

$$𝐗\sim 𝒩(\mathit{\mu },\mathrm{\Sigma })$$

where:

• $\mathit{\mu }=({\mu }_1,{\mu }_2,\dots ,{\mu }_k{)}^T$ is the mean vector.

• $\mathrm{\Sigma }$ is the $k\times k$ covariance matrix, which is symmetric and positive semi-definite.

Probability Density Function

The probability density function (PDF) of the multivariate normal distribution is given by:

$$f(𝐱)=\frac{1}{(2\pi {)}^{k/2}|\mathrm{\Sigma }{|}^{1/2}}\exp (-\frac{1}{2}(𝐱-\mathit{\mu }{)}^T{\mathrm{\Sigma }}^{-1}(𝐱-\mathit{\mu }))$$

for $𝐱\in {ℝ}^k$, where $|\mathrm{\Sigma }|$ denotes the determinant of the covariance matrix.

Properties

1. Marginal Distributions: Any subset of components of a multivariate normal vector is also normally distributed.

2. Conditional Distributions: The conditional distribution of a subset of variables given others is also multivariate normal.

3. Independence: Two components $X_i$ and $X_j$ are independent if and only if the corresponding covariance $\text{Cov}(X_i,X_j)=0$.

Applications

Multivariate normal distributions are widely used in statistics, finance, machine learning, and many fields where multivariate data are analyzed. They are particularly useful for modeling relationships between correlated random variables.