Expected Value and Variance Calculation in IB Mathematics | IB数学：期望值与方差计算详解

📚 Expected Value and Variance Calculation in IB Mathematics | IB数学：期望值与方差计算详解

In IB Mathematics, expected value and variance are two of the most fundamental concepts in probability and statistics. They allow us to summarise an entire probability distribution using just two numbers: a measure of centre and a measure of spread. Mastering these tools is essential for success in both Analysis and Approaches (AA) and Applications and Interpretation (AI) courses.

在IB数学中，期望值和方差是概率与统计中最基本的概念之一。它们让我们能够仅用两个数字来概括整个概率分布：一个衡量中心位置，一个衡量离散程度。掌握这些工具对于分析与方法（AA）和应用与解释（AI）两门课程的成功都至关重要。

1. Random Variables and Probability Distributions | 随机变量与概率分布

A random variable is a variable whose possible values are numerical outcomes of a random experiment. We denote discrete random variables with capital letters, such as X, and their possible values with lowercase letters, such as x.

随机变量是其可能取值为随机试验数值结果的变量。我们用大写字母（如X）表示离散随机变量，用小写字母（如x）表示其可能的取值。

For a discrete random variable, the probability distribution lists each possible value x and its corresponding probability P(X = x). Two conditions must always hold: first, P(X = x) ≥ 0 for every value of x; second, the sum of all probabilities is exactly 1, written as Σ P(X = x) = 1.

对于离散随机变量，概率分布列出每个可能的取值x及其对应的概率P(X = x)。两个条件必须始终成立：第一，每个取值的概率P(X = x) ≥ 0；第二，所有概率之和恰好等于1，记为Σ P(X = x) = 1

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