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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].
It is worth restating the above result in words: the expected value of the score, at true parameter value is zero. Thus, if one were to repeatedly sample from some distribution, and repeatedly calculate the score, then the mean value of the scores would tend to zero asymptotically .
The formula in the definition of characteristic function allows us to compute φ when we know the distribution function F (or density f). If, on the other hand, we know the characteristic function φ and want to find the corresponding distribution function, then one of the following inversion theorems can be used. Theorem.
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
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.
The moment generating function of a real random variable is the expected value of , as a function of the real parameter . For a normal distribution with density f {\textstyle f} , mean μ {\textstyle \mu } and variance σ 2 {\textstyle \sigma ^{2}} , the moment generating function exists and is equal to
Expected value of sample information (EVSI) is a relaxation of the expected value of perfect information (EVPI) metric, which encodes the increase of utility that would be obtained if one were to learn the true underlying state, .
For a Bernoulli random variable, the expected value is the theoretical probability of success, and the average of n such variables (assuming they are independent and identically distributed (i.i.d.)) is precisely the relative frequency. This image illustrates the convergence of relative frequencies to their theoretical probabilities.