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In classical mechanics, the center of mass is an analogous concept to expectation. For example, suppose X is a discrete random variable with values x i and corresponding probabilities p i. Now consider a weightless rod on which are placed weights, at locations x i along the rod and having masses p i (whose sum is one).
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 ...
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
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, a martingale is a sequence of random variables (i.e., a stochastic process) for which, at a particular time, the conditional expectation of the next value in the sequence is equal to the present value, regardless of all prior values. Stopped Brownian motion is an example of a martingale. It can model an even coin-toss ...
Taking the expectation of this conditional variance across all values of X gives [ ()], often termed the “unexplained” or within-group part. The variance of the conditional mean, Var ( E [ Y ∣ X ] ) {\displaystyle \operatorname {Var} (\operatorname {E} [Y\mid X])} , measures how much these conditional means differ (i.e. the ...
For example, if X and Y are uncorrelated and the weight of X is two times the weight of Y, then the weight of the variance of X will be four times the weight of the variance of Y. The expression above can be extended to a weighted sum of multiple variables:
Here [] stands for the expectation conditioned to the σ-algebra . This general statement reduces to the previous ones when the topological vector space T is the real axis , and G {\displaystyle {\mathfrak {G}}} is the trivial σ -algebra {∅, Ω} (where ∅ is the empty set , and Ω is the sample space ).