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Hence ~ gives an estimate of the population variance that is biased by a factor of as the expectation value of ~ is smaller than the population variance (true variance) by that factor. For this reason, S ~ Y 2 {\textstyle {\tilde {S}}_{Y}^{2}} is referred to as the biased sample variance .
Since the quadratic form is a scalar quantity, = (). Next, by the cyclic property of the trace operator, [ ()] = [ ()]. Since the trace operator is a linear combination of the components of the matrix, it therefore follows from the linearity of the expectation operator that
Expected values can also be used to compute the variance, by means of the computational formula for the variance = [] ( []). A very important application of the expectation value is in the field of quantum mechanics.
The conditional variance tells us how much variance is left if we use to "predict" Y. Here, as usual, E ( Y ∣ X ) {\displaystyle \operatorname {E} (Y\mid X)} 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).
In probability theory, the law of total variance [1] or variance decomposition formula or conditional variance formulas or law of iterated variances also known as Eve's law, [2] states that if and are random variables on the same probability space, and the variance of is finite, then
[4]: 29 The general form of the variance function is presented under the exponential family context, as well as specific forms for Normal, Bernoulli, Poisson, and Gamma. In addition, we describe the applications and use of variance functions in maximum likelihood estimation and quasi-likelihood estimation.
which is an unbiased estimator of the variance of the mean in terms of the observed sample variance and known quantities. If the autocorrelations are identically zero, this expression reduces to the well-known result for the variance of the mean for independent data. The effect of the expectation operator in these expressions is that the ...
A much simpler result, stated in a section above, is that the variance of the product of zero-mean independent samples is equal to the product of their variances. Since the variance of each Normal sample is one, the variance of the product is also one. The product of two Gaussian samples is often confused with the product of two Gaussian PDFs.