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The proposition in probability theory known as the law of total expectation, [1] the law of iterated expectations [2] (LIE), Adam's law, [3] the tower rule, [4] and the smoothing theorem, [5] among other names, states that if is a random variable whose expected value is defined, and is any random variable on the same probability space, then
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 law of total covariance can be proved using the law of total expectation: First, (,) = [] [] [] from a simple standard identity on covariances. Then we apply the law of total expectation by conditioning on the random variable Z:
The conditional expectation of rainfall for an otherwise unspecified day known to be (conditional on being) in the month of March, is the average of daily rainfall over all 310 days of the ten–year period that fall in March. Similarly, the conditional expectation of rainfall conditional on days dated March 2 is the average of the rainfall ...
Expectation (or mean), variance and covariance. Jensen's inequality; General moments about the mean; Correlated and uncorrelated random variables; Conditional expectation: law of total expectation, law of total variance; Fatou's lemma and the monotone and dominated convergence theorems; Markov's inequality and Chebyshev's inequality
Conditional probabilities, conditional expectations, and conditional probability distributions are treated on three levels: discrete probabilities, probability density functions, and measure theory. Conditioning leads to a non-random result if the condition is completely specified; otherwise, if the condition is left random, the result of ...
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.
Conditional expectation; Expectation (epistemic) Expectile – related to expectations in a way analogous to that in which quantiles are related to medians; Law of total expectation – the expected value of the conditional expected value of X given Y is the same as the expected value of X; Median – indicated by in a drawing above