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In probability theory, the expected value (also called expectation, expectancy, expectation operator, mathematical expectation, mean, expectation value, or first moment) is a generalization of the weighted average.
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
The unconditional expectation of rainfall for an unspecified day is the average of the rainfall amounts for those 3652 days. 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 ...
An expectation E on an algebra A of random variables is a normalized, positive linear functional. What this means is that E[k] = k where k is a constant; E[X * X] ≥ 0 for all random variables X; E[X + Y] = E[X] + E[Y] for all random variables X and Y; and; E[kX] = kE[X] if k is a constant. One may generalize this setup, allowing the algebra ...
In probability theory, Wald's equation, Wald's identity [1] or Wald's lemma [2] is an important identity that simplifies the calculation of the expected value of the sum of a random number of random quantities.
This proposition is (sometimes) known as the law of the unconscious statistician because of a purported tendency to think of the aforementioned law as the very definition of the expected value of a function g(X) and a random variable X, rather than (more formally) as a consequence of the true definition of expected value. [1]
Similarly, if a submartingale and a martingale have equivalent expectations for a given time, the history of the submartingale tends to be bounded above by the history of the martingale. Roughly speaking, the prefix "sub-" is consistent because the current observation X n is less than (or equal to) the conditional expectation E[X n +1 | X 1 ...
Substituting this estimate in the formula for the expected value of a geometric distribution and solving for gives the estimators ^ = ¯ and ^ = ¯ + when supported on and respectively. These estimators are biased since E ( 1 x ¯ ) > 1 E ( x ¯ ) = p {\displaystyle \mathrm {E} \left({\frac {1}{\bar {x}}}\right)>{\frac {1}{\mathrm {E} ({\bar {x ...