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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 EX of X (sometimes denoted ) is defined as

where the Stieltjes 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 x₁, x₂, ... and corresponding probabilities p₁, p₂, ... which add up to 1, then EX can be computed as the sum or series

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

It follows directly from the discrete case definition that if is a constant random variable, i.e. for some fixed real numberIn mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. The term "real number" is a retronym coined in response to " imaginary number". Real numbers may , then the expected value of is also .

The expected value operator (or expectation operator) E is linear in the sense that

for any two random variables and (which need to be defined on the same probability space) and any real numbers and .

The expected values of the powers of X are called the moments of X; the moments about the meanThe k''th moment about the mean (or k''th central moment of a real-valued random variable X is the quantity E[ X − E X k , where E is the expectation operator. Some random variables have no mean, in which case the moment about the mean is not define of X are expected values of powers of .

In general, the expected value operator is not multiplicative, i.e. E(XY) is not necessarily equal to EX EY, except if X and Y are independentIn probability theory, to say that two events are independent intuitively means that knowing whether or not one of them occurs makes it neither more probable nor less probable that the other occurs. For example, the event of getting a "1" when a die is th or uncorrelatedIn probability theory and statistics, to call two real-valued random variables X and Y uncorrelated means that their correlation is zero, or, equivalently, their covariance is zero. If X and Y are independent then they are uncorrelated. It is not true, ho. The difference, in the general case, gives rise to the covarianceIn probability theory and statistics, the covariance between two real-valued random variables X and Y with expected values and is defined as: : where E is the expectation operator. This is equivalent to the following formula which is commonly used in actu and correlationIn probability theory and statistics, the correlation also called correlation coefficient between two random variables is found by dividing their covariance by the product of their standard deviations. It is defined only if these standard deviations are f.

To empirically estimate the expected value of a random variable, one repeatedly measures values of the variable and computes the arithmetic meanIn mathematics and statistics, the arithmetic mean of a set of numbers is the sum of all the members of the set divided by the number of items in the set. The word set is used perhaps somewhat loosely; for example, the number 3. 8 could occur more than on 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 EX.

See also an inequality on location and scale parameters.