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Concept
Law of total variance
Summary
In probability theory, the law of total variance or variance decomposition formula or conditional variance formulas or law of iterated variances also known as Eve's law, states that if and are random variables on the same probability space, and the variance of is finite, then
In language perhaps better known to statisticians than to probability theorists, the two terms are the "unexplained" and the "explained" components of the variance respectively (cf. fraction of variance unexplained, explained variation). In actuarial science, specifically credibility theory, the first component is called the expected value of the process variance (EVPV) and the second is called the variance of the hypothetical means (VHM). These two components are also the source of the term "Eve's law", from the initials EV VE for "expectation of variance" and "variance of expectation".
Suppose X is a coin flip with the probability of heads being h. Suppose that when X = heads then Y is drawn from a normal distribution with mean μ_h and standard deviation σ_h, and that when X = tails then Y is drawn from normal distribution with mean μ_t and standard deviation σ_t. Then the first, "unexplained" term on the right-hand side of the above formula is the weighted average of the variances, hσ_h^2 + (1 − h)σ_t^2, and the second, "explained" term is the variance of the distribution that gives μ_h with probability h and gives μ_t with probability 1 − h.
There is a general variance decomposition formula for components (see below). For example, with two conditioning random variables:
which follows from the law of total conditional variance:
Note that the conditional expected value is a random variable in its own right, whose value depends on the value of Notice that the conditional expected value of given the is a function of (this is where adherence to the conventional and rigidly case-sensitive notation of probability theory becomes important!). If we write then the random variable is just Similar comments apply to the conditional variance.
Official source
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