“expected value”的概念、定义、翻译、参考文献-科学参考

expected value

Mathematics

The expected value E(X) of a random variable X is a value that gives the mean value of the distribution. For a discrete random variable X, E(X) = ∑p_ix_i, where p_i = Pr(X = x_i). For a continuous random variable X,

$E (X) = \int_{- \infty}^{\infty} x f (x) d x,$

where f is the probability density function of X. The following laws hold:
1. (i) E(aX+bY) = aE(X)+bE(Y).
2. (ii) When X and Y are independent, E(XY) = E(X)E(Y).
3. (iii) The law of total expectation: for a partition A₁, A₂, …, A_n of the sample space
  
  $E (X) = \sum_{k = 1}^{n} E (X | {|A}_{k}) P (A_{k}) .$
The following is a simple example. Let X be the number obtained when a die is thrown. Let x_i = i, for 1 ≤ i ≤ 6. Then $p_{i} = \frac{1}{6}$ , for all i. So $E (X) = \frac{1}{6} \times 1 + \frac{1}{6} \times 2 + \dots+ \frac{1}{6} \times 6 = 3.5$ .

Statistics

The expected value of a random variable X, is denoted by E(X) and may be interpreted as the long-term average value of X. In the case of a discrete random variable, taking values x₁, x₂,… $expected value$ For a continuous random variable with probability density function f, $expected value$ E(X) is often referred to as the expectation of X or as the mean of X. The word ‘expectation’ has been used in this context since the use of ‘expectatio’ by Huygens in his 1657 treatise on the results of playing games of chance: De Ratiociniis in Ludo Aleae.
If g is any function, the expected value of g(X), E[g(X)], is defined by $expected value$ in the case of a discrete random variable, and by $expected value$ in the case of a continuous random variable. Thus $expected value$ The expected value of a random variable may be thought of as the limiting value of the arithmetic mean of the observed values as the sample size increases. It should be noted that in the case of a discrete random variable the expected value is generally not a possible value. For example, when an unbiased coin is tossed once, the expected value of the number of heads is ½.
Some random variables (for example, those with a Cauchy distribution) do not have an expected value. There are also discrete random variables without expected values, though this situation can only arise when the set of possible values is infinite and the sum involved is the sum of a series that does not converge. For example, if , for r = 1, 2,…, where , then the expression for E(X) is , and this series diverges to ∞. For the expected values of sums and products of random variables, see expectation algebra.

Economics

Given the probability distribution P(x) of a discrete random variable X, the expected value, or the (mathematical) expectation of a function f(X) is defined as

$E [f (x)] = \sum P (x) f (x)$

where the sum is taken over all possible values that X can take. For a continuous random variable X with probability distribution function F(x) the expected value of f(X) is defined as

$\int f (x) d F (x)$

where the integral is taken over all possible values of X. The expected value of X is also referred to as the mean of the distribution of X.

单词	expected value
释义	expected value Mathematics The expected value E(X) of a random variable X is a value that gives the mean value of the distribution. For a discrete random variable X, E(X) = ∑p_ix_i, where p_i = Pr(X = x_i). For a continuous random variable X, $E (X) = \int_{- \infty}^{\infty} x f (x) d x,$ where f is the probability density function of X. The following laws hold: (i) E(aX+bY) = aE(X)+bE(Y). (ii) When X and Y are independent, E(XY) = E(X)E(Y). (iii) The law of total expectation: for a partition A₁, A₂, …, A_n of the sample space $E (X) = \sum_{k = 1}^{n} E (X \| {\|A}_{k}) P (A_{k}) .$ The following is a simple example. Let X be the number obtained when a die is thrown. Let x_i = i, for 1 ≤ i ≤ 6. Then $p_{i} = \frac{1}{6}$ , for all i. So $E (X) = \frac{1}{6} \times 1 + \frac{1}{6} \times 2 + \dots+ \frac{1}{6} \times 6 = 3.5$ . Statistics The expected value of a random variable X, is denoted by E(X) and may be interpreted as the long-term average value of X. In the case of a discrete random variable, taking values x₁, x₂,… $expected value$ For a continuous random variable with probability density function f, $expected value$ E(X) is often referred to as the expectation of X or as the mean of X. The word ‘expectation’ has been used in this context since the use of ‘expectatio’ by Huygens in his 1657 treatise on the results of playing games of chance: De Ratiociniis in Ludo Aleae. If g is any function, the expected value of g(X), E[g(X)], is defined by $expected value$ in the case of a discrete random variable, and by $expected value$ in the case of a continuous random variable. Thus $expected value$ The expected value of a random variable may be thought of as the limiting value of the arithmetic mean of the observed values as the sample size increases. It should be noted that in the case of a discrete random variable the expected value is generally not a possible value. For example, when an unbiased coin is tossed once, the expected value of the number of heads is ½. Some random variables (for example, those with a Cauchy distribution) do not have an expected value. There are also discrete random variables without expected values, though this situation can only arise when the set of possible values is infinite and the sum involved is the sum of a series that does not converge. For example, if , for r = 1, 2,…, where , then the expression for E(X) is , and this series diverges to ∞. For the expected values of sums and products of random variables, see expectation algebra. Economics Given the probability distribution P(x) of a discrete random variable X, the expected value, or the (mathematical) expectation of a function f(X) is defined as $E [f (x)] = \sum P (x) f (x)$ where the sum is taken over all possible values that X can take. For a continuous random variable X with probability distribution function F(x) the expected value of f(X) is defined as $\int f (x) d F (x)$ where the integral is taken over all possible values of X. The expected value of X is also referred to as the mean of the distribution of X.
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