6.6:

6.6: 期望值

JoVE Core
Statistics

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JoVE Core Statistics

Expected Value

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6.5: Unusual Results

6.7: Binomial Probability Distribution

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01:15 min

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April 30, 2023

Overview

预期值称为”长期”平均值或平均值。这意味着，在一遍又一遍的实验中，您会期望这个平均值。预期平均值由符号 μ 表示。计算方法如下:

在方程中，x 是一个事件，P（x）是事件发生的概率。

期望值在决策论中具有实际应用。

本文改编自 Openstax，统计学导论，第 4.2 节均值或预期值和标准差。

Transcript

考虑通过掷骰子 100 次获得的概率分布。均值是使用其公式计算的。

随着 n 的增加，平均值会波动，但从这张平均值与试验次数的关系图中可以看出，随着试验次数的增加，平均值逐渐接近恒定值。

随机变量的预期值是样本量增长到无穷大时的平均值。简单来说，它是结果的长期平均值。

因此，它的公式类似于平均值的公式。

期望值的概念在决策论中很有用。如果一个人在轮盘赌的数字 8 上下注 10 美元，则 38 次机会中有 37 次失败，38 次机会中有 1 次获胜。

如果牌桌上的赢钱是 360 美元，那么这个小机会事件的净收益将是 350 美元。

将随机变量的乘积及其概率相加，以获得期望值。

这个数字告诉我们，每下注 10 美元可能会损失 53 美分。

Key Terms and definitions

Expected Value - The long-term average or 'mean' outcome in a random experiment.
Expected Valuation - Monetary value expectation based on statistical analysis.
Event - A specific outcome or combination of outcomes in a random experiment.
Probability of the Event - The chance that a specific outcome will occur.
Decision Theory - Framework for making choices in complex, uncertain scenarios.

Learning Objectives

Define Expected Value – Explain what it is (e.g., expected value).
Contrast Mean vs Expected Value – Explain key differences (e.g., 7.2k views means).
Explore Examples – Describe scenario (e.g., expected value of a probability distribution).
Explain Calculation Process – Describe formula for expected value calculation.
Apply in Context – Discuss relevance in decision theory and statistics.

Questions that this video will help you answer

What is the keyword and how to calculate it (include probability calculations)?
What are the practical applications of expected value?
What is the difference between mean and expected value?

This video is also useful for

Students – Gain a concrete understanding of expected value and its calculation.
Educators – Provides a clear framework for teaching expected value and its statistical uses.
Researchers – Helps in statistical analysis and event prediction in research.
Science Enthusiasts – Offers insights into probabilities and its numerical representation.

Tags

期望值长期平均值平均值概率决策理论 Openstax 入门统计事件 P（x）标准差