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In general '''expectation''' is what is considered the most likely to happen. A less advantageous result gives rise to the
emotion of '''disappointment'''. If something happens that is not at all expected it is a
surprise. See also
anticipation.
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In
probability (and especially
gambling), the '''expected value''' (or '''expectation''') 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. Note that the value itself may not be expected in the general sense, it may be unlikely or even impossible.
For example, an American
roulette wheel has 38 equally possible outcomes. A bet placed on a single number pays 35-to-1 (this means that he is paid 35 times his bet, while also his bet is returned, together he gets 36 times his bet). So the expected value of the profit resulting from a $1 bet on a single number is, considering all 38 possible outcomes: ( -1 × 37/38 ) + ( 35 × 1/38 ), which is about -0.0526. Therefore one expects, on average, to lose over 5 cents for every dollar bet.
In general, if ''X'' is a
random variable defined on a
probability space (Ω, ''P''), then the '''expected value''' E''X'' of ''X'' (sometimes denoted <math>\langle{}X\rangle</math>) is defined as
:<math>\operatorname{E}X = \int_\Omega X\, dP</math>
'''NOTE''' What is the dP term in the integration? Shouldn't this be
:<math>\operatorname{E}X = \int_X x\, P(x)\, dx</math>
where the
Lebesgue integral is employed. Note that not all random variables have an expected value, since the integral may not exist; see
Cauchy distribution for an example. Two variables with the same
probability distribution will have the same expected value, if it is defined.
If ''X'' is a
discrete random variable with values ''x''
1, ''x''
2, ... and corresponding probabilities ''p''
1, ''p''
2, ... which add up to 1, then E''X'' can be computed as the sum or
series
:<math>\operatorname{E}X = \sum_i p_i x_i</math>
as in the gambling example mentioned above.
If the
probability distribution of ''X'' admits a
probability density function ''f''(''x''), then the expected value can be computed as
:<math>\operatorname{E}X = \int_{-\infty}^\infty x f(x)\, dx.</math>
It follows directly from the discrete case definition that if <math>X</math> is a
constant random variable, i.e. <math>X=b</math> for some fixed
real number <math>b</math>, then the expected value of <math>X</math> is also <math>b</math>.
The expected value operator (or '''expectation operator''') E is
linear in the sense that
:<math>\operatorname{E}(aX+Y)=a\operatorname{E}X+\operatorname{E}Y</math>
for any two random variables <math>X</math> and <math>Y</math> (which need to be defined on the same probability space) and any real number <math>a</math>. From this relation, it is trivially obvious that for any two real numbers <math>a</math> and <math>b</math>, and <math>X</math> and <math>Y</math> as before,
:<math>\operatorname{E}(aX+bY)=a\operatorname{E}X+b\operatorname{E}Y</math>
The expected values of the powers of ''X'' are called the ''moments'' of ''X''; the
moments about the mean of ''X'' are also defined as certain expected values.
In general, the expected value operator is not multiplicative, i.e. E(''XY'') is not necessarily equal to E''X'' E''Y'', except if ''X'' and ''Y'' are
independent or
uncorrelated. The difference, in the general case, gives rise to the
covariance and
correlation.
To empirically estimate the expected value of a random variable, one repeatedly measures values of the variable and computes the
arithmetic mean of the results. This estimates the true expected value and has the property of
minimizing the sum of the squares of the errors away from the expected value.
In
classical mechanics, the
center of mass is an analogous concept to expectation. For example, suppose ''X'' is a discrete random variable with values ''x
i'' and corresponding probabilities ''p
i''. Now consider a weightless rod on which are placed weights, at locations ''x
i'' along the rod and having masses ''p
i''. The point at which the rod balances (its
center of gravity) is E''X''.
See also
an inequality on location and scale parameters.
de:Erwartungswert
es:Valor esperado
fr:Esprance mathmatique
nl:Verwachting (wiskunde)
ja:期待値
pl:Wartość oczekiwana
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sv:Vntevrde
