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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.
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 ...
In probability theory and statistics, the law of the unconscious statistician, or LOTUS, is a theorem which expresses the expected value of a function g(X) of a random variable X in terms of g and the probability distribution of X. The form of the law depends on the type of random variable X in question.
The expected return (or expected gain) on a financial investment is the expected value of its return (of the profit on the investment). It is a measure of the center of the distribution of the random variable that is the return. [1] It is calculated by using the following formula: [] = = where
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 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 of how far a set of numbers is spread out from their average value.
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
If the random variable is denoted by , then the mean is also known as the expected value of (denoted ()). For a discrete probability distribution , the mean is given by ∑ x P ( x ) {\displaystyle \textstyle \sum xP(x)} , where the sum is taken over all possible values of the random variable and P ( x ) {\displaystyle P(x)} is the probability ...
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