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Informally, the expected value is the mean of the possible values a random variable can take, weighted by the probability of those outcomes. Since it is obtained through arithmetic, the expected value sometimes may not even be included in the sample data set; it is not the value you would "expect" to get in reality.
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 expected value of g(X) is then identified as (()) ′ = (), where the equality follows by another use of the change-of-variables formula for integration. This shows that the expected value of g ( X ) is encoded entirely by the function g and the density f of X .
In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion, meaning it is a measure
When the model has been estimated over all available data with none held back, the MSPE of the model over the entire population of mostly unobserved data can be estimated as follows.
In probability theory and statistics, the continuous uniform distributions or rectangular distributions are a family of symmetric probability distributions.Such a distribution describes an experiment where there is an arbitrary outcome that lies between certain bounds. [1]
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 ...
In other words, the expected value of the uncorrected sample variance does not equal the population variance σ 2, unless multiplied by a normalization factor. The sample mean, on the other hand, is an unbiased [5] estimator of the population mean μ. [3]