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With any number of random variables in excess of 1, the variables can be stacked into a random vector whose i th element is the i th random variable. Then the variances and covariances can be placed in a covariance matrix, in which the (i, j) element is the covariance between the i th random variable and the j th one.
Use of a user-defined function sq(x) in Microsoft Excel. The named variables x & y are identified in the Name Manager. The function sq is introduced using the Visual Basic editor supplied with Excel. Subroutine in Excel calculates the square of named column variable x read from the spreadsheet, and writes it into the named column variable y.
Note: The conditional expected values E( X | Z) and E( Y | Z) are random variables whose values depend on the value of Z. Note that the conditional expected value of X given the event Z = z is a function of z. If we write E( X | Z = z) = g(z) then the random variable E( X | Z) is g(Z). Similar comments apply to the conditional covariance.
Seen as a function of for given , (= | =) is a probability mass function and so the sum over all (or integral if it is a conditional probability density) is 1. Seen as a function of x {\displaystyle x} for given y {\displaystyle y} , it is a likelihood function , so that the sum (or integral) over all x {\displaystyle x} need not be 1.
In many cases where the principle could give an exact formula (in particular, counting prime numbers using the sieve of Eratosthenes), the formula arising does not offer useful content because the number of terms in it is excessive. If each term individually can be estimated accurately, the accumulation of errors may imply that the inclusion ...
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where (,) is the copula density function, () and () are the marginal probability density functions of X and Y, respectively. There are four elements in this equation, and if any three elements are known, the fourth element can be calculated.
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