| π can be defined by applying an affine transformation π=ππ+b, where b is an π has a probability density function fπ(y)=fπ(πβ1(yβb))|detβ‘π|.
More generally we can study invertible mappings of random vectors.
Let βn onto a subset g have continuous partial derivatives in g be zero at no point of fπ(π±) and satisfies fπ(π²)=fπ(π±)|detβ‘(βπ²βπ±)||π±=gβ1(π²)π(π²βRπ)
where π.
The expected value or mean of a random vector
nΓ1 random vector is an [πβEβ‘[π]][πβEβ‘[π]]T, where the superscript T refers to the transpose of the indicated vector:
\\operatorname{K}_{\\mathbf{X}\\mathbf{X}} = \\operatorname{Var}[\\mathbf{X}]=\\operatorname{E}[(\\mathbf{X}-\\operatorname{E}[\\mathbf{X}])(\\mathbf{X}-\\operatorname{E}[\\mathbf{X}])^{T}] = \\operatorname{E}[\\mathbf{X} \\mathbf{X}^T] - \\operatorname{E}[\\mathbf{X}]\\operatorname{E}[\\mathbf{X}]^T</math>|{{EquationRef|Eq.3}}}}"},"cellpadding":{"wt":"6"},"1":{"wt":"border\n"},"border colour":{"wt":"#0073CF"},"background colour":{"wt":"#F5FFFA"}},"i":0}}]}'>
π and n elements and nΓp matrix
\\operatorname{K}_{\\mathbf{X}\\mathbf{Y}} = \\operatorname{Cov}[\\mathbf{X},\\mathbf{Y}]=\\operatorname{E}[(\\mathbf{X}-\\operatorname{E}[\\mathbf{X}])(\\mathbf{Y}-\\operatorname{E}[\\mathbf{Y}])^{T}] = \\operatorname{E}[\\mathbf{X} \\mathbf{Y}^T] - \\operatorname{E}[\\mathbf{X}]\\operatorname{E}[\\mathbf{Y}]^T</math>|{{EquationRef|Eq.4}}}}"},"cellpadding":{"wt":"6"},"1":{"wt":"border\n"},"border colour":{"wt":"#0073CF"},"background colour":{"wt":"#F5FFFA"}},"i":0}}]}'>
π and the j element of πTKπππβ₯0for allΒ πββn.
The cross-covariance matrix Kππ=KππT.
Two random vectors Eβ‘[ππT]=Eβ‘[π]Eβ‘[π]T.
They are uncorrelated if and only if their cross-covariance matrix nΓn matrix whose (i,j) element is the correlation between the i and the j random variables. The correlation matrix is the expected value, element by element, of the
π and n elements and nΓp matrix
\\operatorname{R}_{\\mathbf{X}\\mathbf{Y}} = \\operatorname{E}[\\mathbf{X} \\mathbf{Y}^T]</math>|{{EquationRef|Eq.6}}}}"},"cellpadding":{"wt":"6"},"1":{"wt":"border\n"},"border colour":{"wt":"#0073CF"},"background colour":{"wt":"#F5FFFA"}},"i":0}}]}'>
Rππ=Kππ+Eβ‘[π]Eβ‘[π]T.
Similarly for the cross-correlation matrix and the cross-covariance matrix:
- π=(Y1,...,Yn)T are called orthogonal if
-
Two random vectors π± and Fπ(π±) and π andπ is often denoted by π are called independent if for all π with π=(Ο1,β¦,Οn)T to a complex number. It is defined by
- Eβ‘[πTAπ]=Eβ‘[π]TAEβ‘[π]+trβ‘(AKππ),
where tr refers to the trace of a matrix β that is, to the sum of the elements on its main diagonal (from upper left to lower right). Since the quadratic form is a scalar, so is its expectation.
Proof: Let Eβ‘[π³]=ΞΌ and mΓm non-stochastic matrix.
Then based on the formula for the covariance, if we denote Covβ‘[π,π]=Eβ‘[ππT]βEβ‘[π]Eβ‘[π]T
Hence
- trβ‘(AB)=trβ‘(BA)).
We see that
- (zβΞΌ)T(AzβAΞΌ)=trβ‘((zβΞΌ)T(AzβAΞΌ))=trβ‘((zβΞΌ)TA(zβΞΌ))
trivially. Using the permutation we get:
- π as follows:
- π. Again, since both quadratic forms are scalars and hence their product is a scalar, the expectation of their product is also a scalar.
In portfolio theory in finance, an objective often is to choose a portfolio of risky assets such that the distribution of the random portfolio return has desirable properties. For example, one might want to choose the portfolio return having the lowest variance for a given expected value. Here the random vector is the vector π«) and the variance of the portfolio return can be shown to be wCw, where C is the covariance matrix of Ξ²Μ is chosen as an estimate of Ξ², and the estimate of the vector e, denoted Ξ²Μ and πt=c+A1πtβ1+A2πtβ2+β―+Apπtβp+πt,
where the i-periods-back vector observation πt is a kΒ ΓΒ 1 random vector of error terms.
Gubner, John A. (2006). Probability and Random Processes for Electrical and Computer Engineers. Cambridge University Press. ISBNΒ 978-0-521-86470-1.
- β Stark, Henry; Woods, John W. (2012). "Random Vectors". Probability, Statistics, and Random Processes for Engineers (FourthΒ ed.). Pearson. pp.Β 295β339. ISBNΒ 978-0-13-231123-6.
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