Prab at 17:07, 22 March 2025

2025-03-22T17:07:16Z

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'''Expected''' is a term often used in [[statistics]] and [[probability theory]] to denote the average outcome of a random event. The expected value is calculated by multiplying each of the possible outcomes by the likelihood each outcome will occur, and then summing all of those values.

==Definition==
In probability theory, the '''expected value''' of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one "expects" to win per bet if bets with identical odds are repeated many times. A game or situation in which the expected value for the player is zero (no net gain nor loss) is called a fair game.

==Calculation==
The expected value is a key aspect of how one characterizes a probability distribution; it is one type of location parameter. By contrast, the variance is a measure of dispersion of the possible values of the random variable around the expected value. The variance itself is defined in terms of two expectations: it is the expected value of the squared deviation of the variable's value from the variable's expected value.

==Applications==
The expected value plays important roles in a variety of contexts. In regression analysis, one desires a formula in terms of observed data that will give a "good" estimate of the parameter giving the effect of some explanatory variable upon a dependent variable. The formula will give different estimates using different samples of data, so the estimate it gives is itself a random variable. A formula is generally considered good in this context if it is an unbiased estimator—that is, if the expected value of the estimate (the average value it would give over an arbitrarily large number of separate samples) can be shown to equal the true value of the desired parameter.

==See also==
* [[Probability theory]]
* [[Statistics]]
* [[Variance]]
* [[Regression analysis]]

[[Category:Statistics]]
[[Category:Probability theory]]
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Prab at 17:07, 22 March 2025

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