Expected Value and Variance | 期望值与方差

📚 Expected Value and Variance | 期望值与方差

Expected value and variance are two of the most fundamental concepts in probability and statistics. They provide a numerical summary of a probability distribution, telling us about the “centre” and “spread” of a random variable. In IB Mathematics, understanding how to compute and interpret E(X) and Var(X) is essential for modelling real-world situations.

期望值和方差是概率与统计学中最基本的概念之一。它们作为概率分布的数值概括，告诉我们随机变量的“中心”和“分散程度”。在IB数学中，理解如何计算和解释 E(X) 和 Var(X) 对于模拟现实情况至关重要。

1. Random Variables Refresher | 随机变量回顾

A random variable X is a function that assigns a real number to each outcome of a random experiment. There are two types: discrete random variables (countable values, often integers) and continuous random variables (any value in an interval). In this article we focus on discrete variables, which are described by a probability distribution table.

随机变量 X 是一个将实数赋值给随机实验每个结果的函数。有两种类型：离散随机变量（可数个值，通常是整数）和连续随机变量（区间内的任何值）。本文重点讨论离散变量，它们由概率分布表描述。

For a discrete random variable, the sum of all probabilities equals 1: ∑ P(X = x) = 1, and each probability satisfies 0 ≤ P(X = x) ≤ 1.

对于离散随机变量，所有概率之和为 1: ∑ P(X = x) = 1，且每个概率满足 0 ≤ P(X = x) ≤ 1。

2. Definition of Expected Value | 期望值定义

The expected value (or mean) of a discrete random variable X, denoted E(X) or μ (mu), is the weighted average of all its possible values, where the weights are the corresponding probabilities.

离散随机变量 X 的期望值（或均值），记作 E(X) 或 μ（mu），是其所有可能值的加权平均，权重为相应的概率。

E(X) = μ = ∑ x · P(X = x)

This sum runs over all possible values of X. The expected value represents the long-run average if the experiment is repeated many times.

该求和遍历 X 的所有可能值。期望值表示如果实验重复多次的长期平均值。

3. Calculating Expected Value: Examples | 期望值计算示例

Example 1: Fair six-sided die. Let X be the number rolled. Possible values: 1,2,3,4,5,6, each with probability 1/6.

示例 1：公平的六面骰子。令 X 为掷出的点数。可能值: 1,2,3,4,5,6，每个概率为 1/6。

E(X) = 1×(1/6) + 2×(1/6) + 3×(1/6) + 4×(1/6) + 5×(1/6) + 6×(1/6) = (1+2+3+4+5+6)/6 = 21/6 = 3.5

E(X) = 1×(1/6) + 2×(1/6) + … = (1+2+3+4+5+6)/6 = 21/6 = 3.5

Notice that the expected value is not a possible outcome. The long-term average roll is 3.5.

注意期望值并不是一个可能的实际结果。长期的平均掷骰点数为 3.5。

Example 2: Discrete distribution table. Consider a random variable Y with the following probability distribution:

示例 2：离散分布表。考虑随机变量 Y 的概率分布如下：

E(Y) = 0(0.2) + 1(0.3) + 2(0.4) + 3(0.1) = 0 + 0.3 + 0.8 + 0.3 = 1.4

E(Y) = 0(0.2) + 1(0.3) + 2(0.4) + 3(0.1) = 1.4

4. Expected Value of a Function of X | X 函数的期望值

We often need the expected value of a function g(X), such as X². The formula is analogous:

我们经常需要计算 X 的函数 g(X) 的期望值，比如 X²。公式类似：

E[g(X)] = ∑ g(x) · P(X = x)

This is particularly useful for finding E(X²), which appears in the variance formula.

这对于求 E(X²) 特别有用，它出现在方差公式中。

For the die example above, E(X²) = (1²+2²+3²+4²+5²+6²)/6 = (1+4+9+16+25+36)/6 = 91/6 ≈ 15.167.

对于上面的骰

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