Multivariate normal distribution

In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional (univariate) normal distribution to higher dimensions. One definition is that a random vector is said to be k-variate normally distributed if every linear combination of its k components has a univariate normal distribution. Its importance derives mainly from the multivariate central limit theorem. The multivariate normal distribution is often used to describe, at least approximately, any set of (possibly) correlated real-valued random variables each of which clusters around a mean value. X   ∼   N ( μ , Σ ) ⟺ there exist  μ ∈ R k , A ∈ R k × ℓ  such that  X = A Z + μ  for  Z n ∼   N ( 0 , 1 ) , i.i.d. {displaystyle mathbf {X} sim {mathcal {N}}(mathbf {mu } ,{oldsymbol {Sigma }})quad iff quad { ext{there exist }}mathbf {mu } in mathbb {R} ^{k},{oldsymbol {A}}in mathbb {R} ^{k imes ell }{ ext{ such that }}mathbf {X} ={oldsymbol {A}}mathbf {Z} +mathbf {mu } { ext{ for }}Z_{n}sim {mathcal {N}}(0,1),{ ext{i.i.d.}}} f X ( x 1 , … , x k ) = exp ⁡ ( − 1 2 ( x − μ ) T Σ − 1 ( x − μ ) ) ( 2 π ) k | Σ | {displaystyle f_{mathbf {X} }(x_{1},ldots ,x_{k})={frac {exp left(-{frac {1}{2}}({mathbf {x} }-{oldsymbol {mu }})^{mathrm {T} }{oldsymbol {Sigma }}^{-1}({mathbf {x} }-{oldsymbol {mu }}) ight)}{sqrt {(2pi )^{k}|{oldsymbol {Sigma }}|}}}} In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional (univariate) normal distribution to higher dimensions. One definition is that a random vector is said to be k-variate normally distributed if every linear combination of its k components has a univariate normal distribution. Its importance derives mainly from the multivariate central limit theorem. The multivariate normal distribution is often used to describe, at least approximately, any set of (possibly) correlated real-valued random variables each of which clusters around a mean value. The multivariate normal distribution of a k-dimensional random vector X = ( X 1 , … , X k ) T {displaystyle mathbf {X} =(X_{1},ldots ,X_{k})^{T}} can be written in the following notation: or to make it explicitly known that X is k-dimensional, with k-dimensional mean vector and k × k {displaystyle k imes k} covariance matrix The inverse of the covariance matrix is called the precision matrix, denoted by Q = Σ − 1 {displaystyle {oldsymbol {Q}}={oldsymbol {Sigma }}^{-1}} . A real random vector X = ( X 1 , … , X k ) T {displaystyle mathbf {X} =(X_{1},ldots ,X_{k})^{mathrm {T} }} is called a standard normal random vector if all of its components X n {displaystyle X_{n}} are independent and each is a zero-mean unit-variance normally distributed random variable, i.e. if X n ∼   N ( 0 , 1 ) {displaystyle X_{n}sim {mathcal {N}}(0,1)} for all n {displaystyle n} .:p. 454 A real random vector X = ( X 1 , … , X k ) T {displaystyle mathbf {X} =(X_{1},ldots ,X_{k})^{mathrm {T} }} is called a centered normal random vector if there exists a deterministic k × ℓ {displaystyle k imes ell } matrix A {displaystyle {oldsymbol {A}}} such that A Z {displaystyle {oldsymbol {A}}mathbf {Z} } has the same distribution as X {displaystyle mathbf {X} } where Z {displaystyle mathbf {Z} } is a standard normal random vector with ℓ {displaystyle ell } components.:p. 454 A real random vector X = ( X 1 , … , X k ) T {displaystyle mathbf {X} =(X_{1},ldots ,X_{k})^{mathrm {T} }} is called a normal random vector if there exists a random ℓ {displaystyle ell } -vector Z {displaystyle mathbf {Z} } , which is a standard normal random vector, a k {displaystyle k} -vector μ {displaystyle mathbf {mu } } , and a k × ℓ {displaystyle k imes ell } matrix A {displaystyle {oldsymbol {A}}} , such that X = A Z + μ {displaystyle mathbf {X} ={oldsymbol {A}}mathbf {Z} +mathbf {mu } } .:p. 454:p. 455