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Any definition of expected value may be extended to define an expected value of a multidimensional random variable, i.e. a random vector X. It is defined component by component, as E[X] i = E[X i]. Similarly, one may define the expected value of a random matrix X with components X ij by E[X] ij = E[X ij].
In probability theory, the conditional expectation, conditional expected value, or conditional mean of a random variable is its expected value evaluated with respect to the conditional probability distribution. If the random variable can take on only a finite number of values, the "conditions" are that the variable can only take on a subset of ...
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
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 event = is a function of (this is where adherence to the conventional and rigidly case-sensitive notation of probability theory becomes important!).
For example, while a given person ... The covariance matrix is the expected value, element by element, of the ... Then based on the formula for the covariance, ...
This shows that the expected value of g(X) is encoded entirely by the function g and the density f of X. [6] The assumption that g is differentiable with nonvanishing derivative, which is necessary for applying the usual change-of-variables formula, excludes many typical cases, such as g(x) = x 2.
If p = 1/n and X is geometrically distributed with parameter p, then the distribution of X/n approaches an exponential distribution with expected value 1 as n → ∞, since (/ >) = (>) = = = [()] [] =. More generally, if p = λ/n, where λ is a parameter, then as n→ ∞ the distribution of X/n approaches an exponential distribution with rate ...
In decision theory, the expected value of sample information (EVSI) is the expected increase in utility that a decision-maker could obtain from gaining access to a sample of additional observations before making a decision.