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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
The simplest estimators for population mean and population variance are simply the mean and variance of the sample, the sample mean and (uncorrected) sample variance – these are consistent estimators (they converge to the value of the whole population as the number of samples increases) but can be improved.
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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.
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The variance of the random variable resulting from an algebraic operation between random variables can be calculated using the following set of rules: Addition : V a r [ Z ] = V a r [ X + Y ] = V a r [ X ] + 2 C o v [ X , Y ] + V a r [ Y ] {\displaystyle \mathrm {Var} [Z]=\mathrm {Var} [X+Y]=\mathrm {Var} [X]+2\mathrm {Cov} [X,Y]+\mathrm {Var ...