Bayes' rule

In mathematics, probability, and statistics, a multivariate random variable or random vector is a list of mathematical variables each of whose value is unknown, either because the value has not yet occurred or because there is imperfect knowledge of its value. The individual variables in a random vector are grouped together because there may be correlations among them — often they represent different properties of an individual statistical unit (e.g. a particular person, event, etc.). Normally each element of a random vector is a real number.

Random vectors are often used as the underlying implementation of various types of aggregate random variables, e.g. a random matrix, random tree, random sequence, random process, etc.

More formally, a multivariate random variable is a column vector $\mathbf{X}=(X_1,...,X_n{)}^T$ (or its transpose, which is a row vector) whose components are scalar-valued random variables on the same probability space $(\mathrm{\Omega },\mathcal{ℱ},P)$, where $\mathrm{\Omega }$ is the sample space, $\mathcal{ℱ}$ is the sigma-algebra (the collection of all events), and $P$ is the probability measure (a function returning each event's probability).

Probability distribution

Every random vector gives rise to a probability measure on ${\mathbb{ℝ}}^n$ with the Borel algebra as the underlying sigma-algebra. This measure is also known as the joint probability distribution, the joint distribution, or the multivariate distribution of the random vector.

The distributions of each of the component random variables $X_i$ are called marginal distributions. The conditional probability distribution of $X_i$ given $X_j$ is the probability distribution of $X_i$ when $X_j$ is known to be a particular value.

Operations on random vectors

Random vectors can be subjected to the same kinds of algebraic operations as can non-random vectors: addition, subtraction, multiplication by a scalar, and the taking of inner products.

Similarly, a new random vector $\mathbf{Y}$ can be defined by applying an affine transformation $g:{\mathbb{ℝ}}^n\to {\mathbb{ℝ}}^n$ to a random vector $\mathbf{X}$:

$\mathbf{Y}=\mathcal{𝒜}\mathbf{X}+b$, where $\mathcal{𝒜}$ is an $n\times n$ matrix and $b$ is an $n\times 1$ column vector.

If $\mathcal{𝒜}$ is invertible and the probability density of $\mathbf{X}$ is $f_{\mathbf{X}}$, then the probability density of $\mathbf{Y}$ is

$f_{\mathbf{Y}(y)}=\frac{f_{\mathbf{X}}({\mathcal{𝒜}}^{-1}(y-b))}{|\det \mathcal{𝒜}|}$.

Expected value, covariance, and cross-covariance

The expected value or mean of a random vector $\mathbf{X}$ is a fixed vector $\mathrm{E}[\mathbf{X}]$ whose elements are the expected values of the respective random variables.

The covariance matrix (also called the variance-covariance matrix) of an $n\times 1$ random vector is an $n\times n$ matrix whose $i,j^{th}$ element is the covariance between the $i^{th}$ and the $j^{th}$ random variables. The covariance matrix is the expected value, element by element, of the $n\times n$ matrix computed as $[\mathbf{X}-\mathrm{E}[\mathbf{X}]][\mathbf{X}-\mathrm{E}[\mathbf{X}]{]}^T$, where the superscript T refers to the transpose of the indicated vector:

$\mathrm{Var}[\mathbf{X}]=\mathrm{E}[(\mathbf{X}-\mathrm{E}[\mathbf{X}])(\mathbf{X}-\mathrm{E}[\mathbf{X}]{)}^T].$

By extension, the cross-covariance matrix between two random vectors $\mathbf{X}$ and $\mathbf{Y}$ ($\mathbf{X}$ having $n$ elements and $\mathbf{Y}$ having $p$ elements) is the $n\times p$ matrix

$\mathrm{Cov}[\mathbf{X},\mathbf{Y}]=\mathrm{E}[(\mathbf{X}-\mathrm{E}[\mathbf{X}])(\mathbf{Y}-\mathrm{E}[\mathbf{Y}]{)}^T],$

where again the indicated matrix expectation is taken element-by-element in the matrix. The cross-covariance matrix $\mathrm{Cov}[\mathbf{Y},\mathbf{X}]$ is simply the transpose of the matrix $\mathrm{Cov}[\mathbf{X},\mathbf{Y}]$.

Further properties

Expectation of a quadratic form

One can take the expectation of a quadratic form in the random vector X as follows:Primarily based on the most recent URA personal property value index (PPPI) flash estimates, we know that the PPPI, which represents the overall real property price development, has dipped in 2013Q4. That is the first dip the market has seen within the final 2 years.

To give you some perspective, the entire number of personal properties in Singapore (together with govt condominiums) is 297,689 in 2013Q3. Primarily based on the projection that there will be 19,302 units accomplished in 2014, the rise in residential models works out to be more than 6%. With a lot New Ec Launch Singapore provide, buyers might be spoilt for alternative and this in flip will lead to their reluctance to pay a premium for potential models. The complete textual content of the Copyright Act (Cap sixty three) and different statutes regarding IPR might be found on the Singapore Statutes Online Website online The Group's income jumped forty.1 p.c to $324.5 million from $231.6 million in FY 2013, lifted by increased development income and sales of growth properties in Singapore and China. Actual Estate Shopping for

One factor we've on this nation is a big group of "economists," and "market analysts." What's interesting about this group of real property market-watchers is that there are two very other ways wherein they predict Boomers will affect housing markets over the subsequent decade. Let's check out those two opposites and see how every can change the best way real property investors strategy their markets. The good news is that actual property buyers are prepared for either state of affairs, and there's profit in being ready. I'm excited and searching ahead to the alternatives both or each of these conditions will supply; thank you Boomers! Mapletree to further broaden past Asia Why fortune will favour the brave in Asia's closing real property frontier

The story of the 23.2 home begins with a stack of Douglas fir beams salvaged from varied demolished warehouses owned by the consumer's household for a number of generations. Design and structure innovator Omer Arbel, configured them to type a triangulated roof, which makes up one of the placing features of the home. The transient from the entrepreneur-proprietor was not solely to design a house that integrates antique wood beams, however one which erases the excellence between inside and exterior. Built on a gentle slope on a large rural acreage surrounded by two masses of previous-development forests, the indoors movement seamlessly to the outdoors and, from the within looking, one enjoys unobstructed views of the existing natural panorama which is preserved

First, there are typically extra rental transactions than gross sales transactions, to permit AV to be decided for each property primarily based on comparable properties. Second, movements in sale costs are more unstable than leases. Hence, utilizing rental transactions to derive the AV helps to maintain property tax more steady for property homeowners. If you're buying or trying to lease a property. It's tiring to call up individual property agent, organize appointments, coordinate timing and to go for particular person property viewing. What most individuals do is to have a property agent representing them who will arrange and coordinate the viewings for all the properties out there based mostly on your necessities & most well-liked timing. Rent Property District 12 Rent Property District thirteen

The Annual Worth of a property is mostly derived based mostly on the estimated annual hire that it may well fetch if it have been rented out. In determining the Annual Worth of a property, IRAS will think about the leases of similar properties within the vicinity, dimension and condition of the property, and different relevant components. The Annual Worth of a property is determined in the identical method regardless of whether the property is let-out, proprietor-occupied or vacant. The Annual Worth of land is determined at 5% of the market price of the land. When a constructing is demolished, the Annual Worth of the land is assessed by this method. Property Tax on Residential Properties Buyer Stamp Responsibility on Buy of Properties – Business and residential properties Rent House District 01

Within the event the Bank's valuation is decrease than the acquisition price, the purchaser has to pay the distinction between the purchase value and the Bank's valuation utilizing money. As such, the money required up-front might be increased so it's at all times essential to know the valuation of the property before making any offer. Appoint Lawyer The Bank will prepare for a proper valuation of the property by way of physical inspection The completion statement will present you the balance of the acquisition price that you must pay after deducting any deposit, pro-rated property tax and utility costs, upkeep prices, and different relevant expenses in addition to any fees payable to the agent and the lawyer. Stamp Responsibility Primarily based on the Purchase Price or Market Value, whichever is larger

$\mathrm{E}(X^{T}AX)=[\mathrm{E}(X){]}^{T}A[\mathrm{E}(X)]+\mathrm{tr}(AC),$

where C is the covariance matrix of X and tr refers to the trace of a matrix — 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 $\mathbf{z}$ be an $m\times 1$ random vector with $\mathrm{E}[\mathbf{z}]=\mu$ and $\mathrm{Cov}[\mathbf{z}]=V$ and let $A$ be an $m\times m$ non-stochastic matrix.

Based on the formula of the covariance, then if we call ${\mathbf{z}}^{\prime }=\mathbf{X}$ and ${\mathbf{z}}^{\prime }{A}^{\prime }=\mathbf{Y}$, we see that:

$\mathrm{Cov}[\mathbf{X},\mathbf{Y}]=\mathrm{E}[\mathbf{X}{\mathbf{Y}}^{\prime }]-\mathrm{E}[\mathbf{X}]\mathrm{E}[\mathbf{Y}{]}^{\prime }$

Hence

$\begin{array}{rl}E(X{Y}^{\prime })& =\mathrm{Cov}(X,Y)+E(X)E(Y{)}^{\prime }\\ E(z^{\prime }Az)& =\mathrm{Cov}(z^{\prime },z^{\prime }{A}^{\prime })+E(z^{\prime })E(z^{\prime }{A}^{\prime }{)}^{\prime }\\ & =\mathrm{Cov}(z^{\prime },z^{\prime }{A}^{\prime })+{\mu }^{\prime }({\mu }^{\prime }{A}^{\prime }{)}^{\prime }\\ & =\mathrm{Cov}(z^{\prime },z^{\prime }{A}^{\prime })+{\mu }^{\prime }A\mu ,\end{array}$

which leaves us to show that

$\mathrm{Cov}(z^{\prime },z^{\prime }{A}^{\prime })=\mathrm{t}(AV).$

This is true based on the fact that one can cyclically permute matrices when taking a trace without changing the end result (e.g.: trace(AB) = trace(BA)).

We see that

$\begin{array}{rl}\mathrm{Cov}(z^{\prime },z^{\prime }{A}^{\prime })& =E\left[(z^{\prime }-E(z^{\prime })){(z^{\prime }{A}^{\prime }-E\left(z^{\prime }{A}^{\prime }\right))}^{\prime }\right]\\ & =E[(z^{\prime }-{\mu }^{\prime })(z^{\prime }{A}^{\prime }-{\mu }^{\prime }{A}^{\prime }{)}^{\prime }]\\ & =E[(z-\mu {)}^{\prime }(Az-A\mu )].\end{array}$

And since

${\left({z^{\prime }-\mu }^{\prime }\right)}^{\prime }\left(Az-A\mu \right)$

is a fixed number, then

$(z-\mu {)}^{\prime }(Az-A\mu )=\mathrm{trace}\left[(z-\mu {)}^{\prime }(Az-A\mu )\right]=\mathrm{trace}[(z-\mu {)}^{\prime }A(z-\mu )]$

trivially. Using the permutation we get:

$\mathrm{trace}\left[(z-\mu {)}^{\prime }A(z-\mu )\right]=\mathrm{trace}\left[A(z-\mu {)}^{\prime }(z-\mu )\right],$

and by plugging this into the original formula we get:

$\begin{array}{rl}\mathrm{Cov}\left(z^{\prime },z^{\prime }{A}^{\prime }\right)& =E\left[{\left({z^{\prime }-\mu }^{\prime }\right)}^{\prime }(Az-A\mu )\right]\\ & =E[\mathrm{trace}[A(z-\mu {)}^{\prime }(z-\mu )]]\\ & =\mathrm{trace}\left[A\cdot E[(z-\mu {)}^{\prime }(z-\mu )]\right]\\ & =\mathrm{trace}[AV].\end{array}$

Expectation of the product of two different quadratic forms

One can take the expectation of the product of two different quadratic forms in a zero-mean Gaussian random vector X as follows:Primarily based on the most recent URA personal property value index (PPPI) flash estimates, we know that the PPPI, which represents the overall real property price development, has dipped in 2013Q4. That is the first dip the market has seen within the final 2 years.

To give you some perspective, the entire number of personal properties in Singapore (together with govt condominiums) is 297,689 in 2013Q3. Primarily based on the projection that there will be 19,302 units accomplished in 2014, the rise in residential models works out to be more than 6%. With a lot New Ec Launch Singapore provide, buyers might be spoilt for alternative and this in flip will lead to their reluctance to pay a premium for potential models. The complete textual content of the Copyright Act (Cap sixty three) and different statutes regarding IPR might be found on the Singapore Statutes Online Website online The Group's income jumped forty.1 p.c to $324.5 million from $231.6 million in FY 2013, lifted by increased development income and sales of growth properties in Singapore and China. Actual Estate Shopping for

One factor we've on this nation is a big group of "economists," and "market analysts." What's interesting about this group of real property market-watchers is that there are two very other ways wherein they predict Boomers will affect housing markets over the subsequent decade. Let's check out those two opposites and see how every can change the best way real property investors strategy their markets. The good news is that actual property buyers are prepared for either state of affairs, and there's profit in being ready. I'm excited and searching ahead to the alternatives both or each of these conditions will supply; thank you Boomers! Mapletree to further broaden past Asia Why fortune will favour the brave in Asia's closing real property frontier

The story of the 23.2 home begins with a stack of Douglas fir beams salvaged from varied demolished warehouses owned by the consumer's household for a number of generations. Design and structure innovator Omer Arbel, configured them to type a triangulated roof, which makes up one of the placing features of the home. The transient from the entrepreneur-proprietor was not solely to design a house that integrates antique wood beams, however one which erases the excellence between inside and exterior. Built on a gentle slope on a large rural acreage surrounded by two masses of previous-development forests, the indoors movement seamlessly to the outdoors and, from the within looking, one enjoys unobstructed views of the existing natural panorama which is preserved

First, there are typically extra rental transactions than gross sales transactions, to permit AV to be decided for each property primarily based on comparable properties. Second, movements in sale costs are more unstable than leases. Hence, utilizing rental transactions to derive the AV helps to maintain property tax more steady for property homeowners. If you're buying or trying to lease a property. It's tiring to call up individual property agent, organize appointments, coordinate timing and to go for particular person property viewing. What most individuals do is to have a property agent representing them who will arrange and coordinate the viewings for all the properties out there based mostly on your necessities & most well-liked timing. Rent Property District 12 Rent Property District thirteen

The Annual Worth of a property is mostly derived based mostly on the estimated annual hire that it may well fetch if it have been rented out. In determining the Annual Worth of a property, IRAS will think about the leases of similar properties within the vicinity, dimension and condition of the property, and different relevant components. The Annual Worth of a property is determined in the identical method regardless of whether the property is let-out, proprietor-occupied or vacant. The Annual Worth of land is determined at 5% of the market price of the land. When a constructing is demolished, the Annual Worth of the land is assessed by this method. Property Tax on Residential Properties Buyer Stamp Responsibility on Buy of Properties – Business and residential properties Rent House District 01

Within the event the Bank's valuation is decrease than the acquisition price, the purchaser has to pay the distinction between the purchase value and the Bank's valuation utilizing money. As such, the money required up-front might be increased so it's at all times essential to know the valuation of the property before making any offer. Appoint Lawyer The Bank will prepare for a proper valuation of the property by way of physical inspection The completion statement will present you the balance of the acquisition price that you must pay after deducting any deposit, pro-rated property tax and utility costs, upkeep prices, and different relevant expenses in addition to any fees payable to the agent and the lawyer. Stamp Responsibility Primarily based on the Purchase Price or Market Value, whichever is larger

$\mathrm{E}[X^{T}AX][X^{T}BX]=2\mathrm{trace}(ACBC)+\mathrm{trace}(AC)\mathrm{trace}(BC)$

where again C is the covariance matrix of X. Again, since both quadratic forms are scalars and hence their product is a scalar, the expectation of their product is also a scalar.

Applications

Portfolio theory

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 r of random returns on the individual assets, and the portfolio return p (a random scalar) is the inner product of the vector of random returns with a vector w of portfolio weights — the fractions of the portfolio placed in the respective assets. Since p = w^Tr, the expected value of the portfolio return is w^TE(r) and the variance of the portfolio return can be shown to be w^TCw, where C is the covariance matrix of r.

Regression theory

In linear regression theory, we have data on n observations on a dependent variable y and n observations on each of k independent variables x_j. The observations on the dependent variable are stacked into a column vector y; the observations on each independent variable are also stacked into column vectors, and these latter column vectors are combined into a matrix X of observations on the independent variables. Then the following regression equation is postulated as a description of the process that generated the data:

$y=X\beta +e,$

where β is a postulated fixed but unknown vector of k response coefficients, and e is an unknown random vector reflecting random influences on the dependent variable. By some chosen technique such as ordinary least squares, a vector $\hat{\beta }$ is chosen as an estimate of β, and the estimate of the vector e, denoted $\hat{e}$, is computed as

$\hat{e}=y-X\hat{\beta }.$

Then the statistician must analyze the properties of $\hat{\beta }$ and $\hat{e}$, which are viewed as random vectors since a randomly different selection of n cases to observe would have resulted in different values for them.

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.