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This implies that in a weighted sum of variables, the variable with the largest weight will have a disproportionally large weight in the variance of the total. For example, if X and Y are uncorrelated and the weight of X is two times the weight of Y, then the weight of the variance of X will be four times the weight of the variance of Y.
Here, as usual, stands for the conditional expectation of Y given X, which we may recall, is a random variable itself (a function of X, determined up to probability one). As a result, Var ( Y ∣ X ) {\displaystyle \operatorname {Var} (Y\mid X)} itself is a random variable (and is a function of X ).
The following formula shows how to apply the general, measure theoretic variance decomposition formula [4] to stochastic dynamic systems. Let Y ( t ) {\displaystyle Y(t)} be the value of a system variable at time t . {\displaystyle t.}
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
Non-parametric estimation of the variance function and its importance, has been discussed widely in the literature [5] [6] [7] In non-parametric regression analysis, the goal is to express the expected value of your response variable(y) as a function of your predictors (X).
A VAR model describes the evolution of a set of k variables, called endogenous variables, over time. Each period of time is numbered, t = 1, ..., T. The variables are collected in a vector, y t, which is of length k. (Equivalently, this vector might be described as a (k × 1)-matrix.) The vector is modelled as a linear function of its previous ...
In the event that the variables X and Y are jointly normally distributed random variables, then X + Y is still normally distributed (see Multivariate normal distribution) and the mean is the sum of the means. However, the variances are not additive due to the correlation. Indeed,
This algorithm can easily be adapted to compute the variance of a finite population: simply divide by n instead of n − 1 on the last line.. Because SumSq and (Sum×Sum)/n can be very similar numbers, cancellation can lead to the precision of the result to be much less than the inherent precision of the floating-point arithmetic used to perform the computation.