Search results
Results From The WOW.Com Content Network
In words, this equation says that the residual is orthogonal to the space M of all functions of Y. This orthogonality condition, applied to the indicator functions f ( Y ) = 1 Y ∈ H {\displaystyle f(Y)=1_{Y\in H}} , is used below to extend conditional expectation to the case that X and Y are not necessarily in L 2 {\displaystyle L^{2}} .
Note that both x and y are evaluated before the results of one or the other are returned from the function. Here, x is returned if the condition holds true and y otherwise. Fortran-2023 has added conditional expressions which evaluate one or the other of the expressions based on the conditional expression:
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 the case of a time series which is stationary in the wide sense, both the means and variances are constant over time (E(X n+m) = E(X n) = μ X and var(X n+m) = var(X n) and likewise for the variable Y). In this case the cross-covariance and cross-correlation are functions of the time difference: cross-covariance
In words: the variance of Y is the sum of the expected conditional variance of Y given X and the variance of the conditional expectation of Y given X. The first term captures the variation left after "using X to predict Y", while the second term captures the variation due to the mean of the prediction of Y due to the randomness of X.
Let and be a continuous random variables with a joint probability density function (,). The differential conditional entropy (|) is defined as [3] ...
Let X be a discrete random variable and its possible outcomes denoted V. For example, if X represents the value of a rolled dice then V is the set {,,,,,}. Let us assume for the sake of presentation that X is a discrete random variable, so that each value in V has a nonzero probability.