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Probability theory. In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is. The parameter is the mean or expectation of the distribution (and also its median and mode ), while ...
Binomial distribution for = with n and k as in Pascal's triangleThe probability that a ball in a Galton box with 8 layers (n = 8) ends up in the central bin (k = 4) is /. In probability theory and statistics, the binomial distribution with parameters n and p is the discrete probability distribution of the number of successes in a sequence of n independent experiments, each asking a yes–no ...
Fibonacci sequence. A tiling with squares whose side lengths are successive Fibonacci numbers: 1, 1, 2, 3, 5, 8, 13 and 21. In mathematics, the Fibonacci sequence is a sequence in which each number is the sum of the two preceding ones. Numbers that are part of the Fibonacci sequence are known as Fibonacci numbers, commonly denoted Fn .
- Mathematical Definition
- Examples
- Properties
- Calculating The Sample Covariance
- Generalizations
- Numerical Computation
- Comments
- Applications
For two jointly distributed real-valued random variables X {\displaystyle X} and Y {\displaystyle Y} with finite second moments, the covariance is defined as the expected value (or mean) of the product of their deviations from their individual expected values:: 119 where E [ X ] {\displaystyle \operatorname {E} [X]} is the expected value of X {\d...
Consider 3 independent random variables A , B , C {\displaystyle A,B,C} and two constants q , r {\displaystyle q,r} . Suppose that X {\displaystyle X} and Y {\displaystyle Y} have the following joint probability mass function, in which the six central cells give the discrete joint probabilities f ( x , y ) {\displaystyle f(x,y)} of the six hypothet...
Covariance with itself
The variance is a special case of the covariance in which the two variables are identical (that is, in which one variable has the same distribution as the other):: 121
Covariance of linear combinations
If X {\displaystyle X} , Y {\displaystyle Y} , W {\displaystyle W} , and V {\displaystyle V} are real-valued random variables and a , b , c , d {\displaystyle a,b,c,d} are real-valued constants, then the following facts are a consequence of the definition of covariance: For a sequence X 1 , … , X n {\displaystyle X_{1},\ldots ,X_{n}} of random variables in real-valued, and constants a 1 , … , a n {\displaystyle a_{1},\ldots ,a_{n}} , we have
Hoeffding's covariance identity
A useful identity to compute the covariance between two random variables X , Y {\displaystyle X,Y} is the Hoeffding's covariance identity:
The sample covariances among K {\displaystyle K} variables based on N {\displaystyle N} observations of each, drawn from an otherwise unobserved population, are given by the K × K {\displaystyle K\times K} matrix q ¯ = [ q j k ] {\displaystyle \textstyle {\overline {\mathbf {q} }}=\left[q_{jk}\right]} with the entries 1. q j k = 1 N − 1 ∑ i = 1 N (...
Auto-covariance matrix of real random vectors
For a vector X = [ X 1 X 2 … X m ] T {\displaystyle \mathbf {X} ={\begin{bmatrix}X_{1}&X_{2}&\dots &X_{m}\end{bmatrix}}^{\mathrm {T} }} of m {\displaystyle m} jointly distributed random variables with finite second moments, its auto-covariance matrix (also known as the variance–covariance matrix or simply the covariance matrix) K X X {\displaystyle \operatorname {K} _{\mathbf {X} \mathbf {X} }} (also denoted by Σ ( X ) {\displaystyle \Sigma (\mathbf {X} )} or cov ( X , X ) {\displaystyle \o...
Cross-covariance matrix of real random vectors
For real random vectors X ∈ R m {\displaystyle \mathbf {X} \in \mathbb {R} ^{m}} and Y ∈ R n {\displaystyle \mathbf {Y} \in \mathbb {R} ^{n}} , the m × n {\displaystyle m\times n} cross-covariance matrix is equal to: 336 where Y T {\displaystyle \mathbf {Y} ^{\mathrm {T} }} is the transpose of the vector (or matrix) Y {\displaystyle \mathbf {Y} } . The ( i , j ) {\displaystyle (i,j)} -th element of this matrix is equal to the covariance cov ( X i , Y j ) {\displaystyle \operatorname {cov} (...
Cross-covariance sesquilinear form of random vectors in a real or complex Hilbert space
More generally let H 1 = ( H 1 , ⟨ , ⟩ 1 ) {\displaystyle H_{1}=(H_{1},\langle \,,\rangle _{1})} and H 2 = ( H 2 , ⟨ , ⟩ 2 ) {\displaystyle H_{2}=(H_{2},\langle \,,\rangle _{2})} , be Hilbert spaces over R {\displaystyle \mathbb {R} } or C {\displaystyle \mathbb {C} } with ⟨ , ⟩ {\displaystyle \langle \,,\rangle } anti linear in the first variable, and let X , Y {\displaystyle \mathbf {X} ,\mathbf {Y} } be H 1 {\displaystyle H_{1}} resp. H 2 {\displaystyle H_{2}} valued random variables. Then...
When E [ X Y ] ≈ E [ X ] E [ Y ] {\displaystyle \operatorname {E} [XY]\approx \operatorname {E} [X]\operatorname {E} [Y]} , the equation cov ( X , Y ) = E [ X Y ] − E [ X ] E [ Y ] {\displaystyle \operatorname {cov} (X,Y)=\operatorname {E} \left[XY\right]-\operatorname {E} \left[X\right]\operatorname {E} \left[Y\right]} is prone to ca...
The covariance is sometimes called a measure of "linear dependence" between the two random variables. That does not mean the same thing as in the context of linear algebra (see linear dependence). When the covariance is normalized, one obtains the Pearson correlation coefficient, which gives the goodness of the fit for the best possible linear func...
In genetics and molecular biology
Covariance is an important measure in biology. Certain sequences of DNA are conserved more than others among species, and thus to study secondary and tertiary structures of proteins, or of RNA structures, sequences are compared in closely related species. If sequence changes are found or no changes at all are found in noncoding RNA (such as microRNA), sequences are found to be necessary for common structural motifs, such as an RNA loop. In genetics, covariance serves a basis for computation o...
In financial economics
Covariances play a key role in financial economics, especially in modern portfolio theory and in the capital asset pricing model. Covariances among various assets' returns are used to determine, under certain assumptions, the relative amounts of different assets that investors should (in a normative analysis) or are predicted to (in a positive analysis) choose to hold in a context of diversification.
In meteorological and oceanographic data assimilation
The covariance matrix is important in estimating the initial conditions required for running weather forecast models, a procedure known as data assimilation. The 'forecast error covariance matrix' is typically constructed between perturbations around a mean state (either a climatological or ensemble mean). The 'observation error covariance matrix' is constructed to represent the magnitude of combined observational errors (on the diagonal) and the correlated errors between measurements (off th...
the free encyclopedia that anyone can edit. Leucippus was a Greek philosopher of the 5th century BCE. He is credited with founding atomism, with his student Democritus. Leucippus divided the world into two entities: atoms, indivisible particles that make up all things, and the void, the nothingness between the atoms.
On 3 April 2024, at 07:58:11 NST (23:58:11 UTC on 2 April), a Mw 7.4 earthquake struck 16 km (9.9 mi)[3] south-southwest of Hualien City, Hualien County, Taiwan. At least 18 people were killed and over 1,100 were injured in the earthquake. It is the strongest earthquake in Taiwan since the 1999 Jiji earthquake,[4] with three aftershocks above ...
Nuclear fusion is a reaction in which two or more atomic nuclei, usually deuterium and tritium (hydrogen isotopes ), combine to form one or more different atomic nuclei and subatomic particles ( neutrons or protons ). The difference in mass between the reactants and products is manifested as either the release or absorption of energy.
