mathematical expectation

简明释义

数学期望值

英英释义

例句

1.Investors often calculate the mathematical expectation (数学期望) of returns to assess the risk and reward of their portfolios.

投资者通常计算回报的数学期望 (数学期望) 来评估其投资组合的风险和回报。

2.When playing the lottery, understanding the mathematical expectation (数学期望) can help you realize the low chances of winning.

在买彩票时，理解数学期望 (数学期望) 可以帮助你意识到获胜的低概率。

3.The mathematical expectation (数学期望) of a random variable gives us the average outcome we can expect from an experiment.

随机变量的数学期望 (数学期望) 为我们提供了我们可以期待的实验平均结果。

4.In gambling, the concept of mathematical expectation (数学期望) helps players understand their potential winnings over time.

在赌博中，数学期望 (数学期望) 的概念帮助玩家理解他们可能的长期收益。

5.The mathematical expectation (数学期望) is crucial in decision-making processes, especially in economics.

在决策过程中，数学期望 (数学期望) 在经济学中尤为重要。

作文

In the realm of statistics and probability theory, the concept of mathematical expectation plays a crucial role in understanding random variables and their behaviors. Essentially, mathematical expectation (or expected value) is a measure of the center of a probability distribution, providing a single value that summarizes the average outcome of a random process. To grasp this concept more thoroughly, we can delve into its definition, significance, and applications in various fields.The formal definition of mathematical expectation is as follows: for a discrete random variable, it is calculated by multiplying each possible outcome by its probability and summing these products. For example, consider a simple game where you roll a fair six-sided die. The outcomes are the numbers 1 through 6, each with a probability of 1/6. The mathematical expectation can be computed as:E(X) = (1 * 1/6) + (2 * 1/6) + (3 * 1/6) + (4 * 1/6) + (5 * 1/6) + (6 * 1/6) = 3.5.This means that if you were to roll the die many times, the average result would approach 3.5. This example illustrates how mathematical expectation provides a way to predict long-term outcomes based on probabilities.The significance of mathematical expectation extends beyond simple games of chance. In finance, for instance, investors use expected value calculations to assess the potential profitability of different investment options. By weighing the expected returns against the risks, they can make informed decisions that align with their financial goals. Similarly, in insurance, companies rely on mathematical expectation to determine premiums based on the likelihood of claims, ensuring they remain profitable while providing coverage to policyholders.Moreover, mathematical expectation is not limited to discrete random variables; it also applies to continuous random variables. In this case, the expected value is calculated using integrals, which allows for a broader range of applications in fields such as economics, engineering, and natural sciences. For example, in economics, the expected utility theory uses mathematical expectation to analyze consumer choices under uncertainty, helping economists understand how individuals make decisions when faced with risk.Despite its wide-ranging applications, mathematical expectation has its limitations. One major drawback is that it does not account for the variability or dispersion of outcomes. Two different random variables can have the same expected value but vastly different distributions. For instance, consider two games: in one game, you win $100 with a probability of 0.5 and lose $0 with a probability of 0.5, while in the other, you win $200 with a probability of 0.25 and lose $100 with a probability of 0.75. Both games have the same mathematical expectation of $50, yet the risk associated with the second game is much higher. Thus, while expected value is a useful tool, it should be complemented with other measures, such as variance and standard deviation, to provide a more comprehensive view of risk.In conclusion, the concept of mathematical expectation is fundamental in the study of probability and statistics. It serves as a powerful tool for summarizing the average outcomes of random processes and is widely applicable in various fields, including finance, insurance, and economics. However, it is essential to recognize its limitations and consider additional statistical measures to gain a complete understanding of risks and uncertainties. As we continue to navigate an increasingly complex world, the ability to calculate and interpret mathematical expectation will remain an invaluable skill for decision-making and analysis.

在统计学和概率论的领域中，数学期望的概念在理解随机变量及其行为方面起着至关重要的作用。基本上，数学期望（或期望值）是概率分布中心的一个度量，提供了一个总结随机过程平均结果的单一值。为了更透彻地掌握这一概念，我们可以深入探讨其定义、重要性及其在各个领域的应用。数学期望的正式定义如下：对于离散随机变量，它是通过将每个可能的结果乘以其概率并对这些乘积求和来计算的。例如，考虑一个简单的游戏，你掷一个公平的六面骰子。结果是1到6的数字，每个数字的概率为1/6。数学期望可以计算为：E(X) = (1 * 1/6) + (2 * 1/6) + (3 * 1/6) + (4 * 1/6) + (5 * 1/6) + (6 * 1/6) = 3.5。这意味着如果你多次掷骰子，平均结果将接近3.5。这个例子说明了数学期望如何提供一种基于概率预测长期结果的方法。数学期望的重要性超越了简单的机会游戏。例如，在金融领域，投资者使用期望值计算来评估不同投资选项的潜在盈利能力。通过权衡预期回报与风险，他们可以做出符合自身财务目标的明智决策。同样，在保险行业，公司依赖数学期望来根据索赔的可能性确定保费，确保在为投保人提供保障的同时保持盈利。此外，数学期望不仅限于离散随机变量；它也适用于连续随机变量。在这种情况下，期望值是通过积分计算的，这使得它在经济学、工程学和自然科学等领域具有更广泛的应用。例如，在经济学中，预期效用理论利用数学期望分析消费者在不确定性下的选择，帮助经济学家理解个人在面对风险时如何做出决策。尽管数学期望有广泛的应用，但它也有其局限性。一个主要缺点是它没有考虑结果的变异性或离散性。两个不同的随机变量可以具有相同的期望值，但分布却大相径庭。例如，考虑两个游戏：在一个游戏中，你以0.5的概率赢得100美元，以0.5的概率输掉0美元，而在另一个游戏中，你以0.25的概率赢得200美元，以0.75的概率输掉100美元。这两个游戏的数学期望都是50美元，但第二个游戏的风险要高得多。因此，虽然期望值是一个有用的工具，但应结合其他度量，如方差和标准差，以提供对风险的更全面的看法。总之，数学期望的概念在概率和统计的研究中是基础性的。它作为一个强大的工具，用于总结随机过程的平均结果，并在金融、保险和经济学等多个领域广泛应用。然而，重要的是要认识到它的局限性，并考虑额外的统计度量，以获得对风险和不确定性的完整理解。随着我们继续在一个日益复杂的世界中航行，计算和解释数学期望的能力将仍然是一项宝贵的决策和分析技能。