Expected Value and Variance of a Function of X | 随机变量函数的期望与方差

📚 Expected Value and Variance of a Function of X | 随机变量函数的期望与方差

For a discrete random variable X, you often need to calculate the mean and spread not only of X itself but also of a new random variable formed by applying a function, such as Y = aX + b or Y = X². This article explains how E[g(X)] and Var[g(X)] are computed, with special attention to linear transformations.

对于离散型随机变量 X，我们常常不仅需要计算 X 本身的均值与离散程度，还要计算由函数产生的新随机变量的期望与方差，例如 Y = aX + b 或 Y = X²。本文讲解如何计算 E[g(X)] 与 Var[g(X)]，并重点关注线性变换。

1. Discrete Random Variables and Expectation | 离散随机变量与期望

A discrete random variable X has a list of possible values x₁, x₂, … and corresponding probabilities P(X = xᵢ). The probabilities must all be non-negative and sum to 1.

离散型随机变量 X 有一组可能取值 x₁、x₂、… 以及相应的概率 P(X = xᵢ)。所有概率都必须是非负的，并且总和等于 1。

The expected value, also called the mean, is written E(X) or μ. It is a probability-weighted average, not necessarily a value that X can actually take.

期望值又称均值，记作 E(X) 或 μ。它是按概率加权的平均值，不一定是 X 实际能取到的值。

E(X) = Σ x · P(X = x)

In Edexcel questions you are usually given a probability distribution table. Always check that the probabilities sum to 1 before calculating E(X).

在 Edexcel 考题中，通常会给出概率分布表。计算 E(X) 之前，务必先检查各概率之和是否等于 1。

2. Variance and Standard Deviation | 方差与标准差

Variance measures how far the values of X are spread from the mean μ = E(X). A small variance means values are concentrated near μ; a large variance means they are more spread out.

方差衡量 X 的取值相对于均值 μ = E(X) 的分散程度。方差小表示取值集中在 μ 附近；方差大表示取值更加分散。

The definition is the expected squared deviation, but the faster formula Var(X) = E(X²) − [E(X)]² is usually used in exam calculations.

定义是偏差平方的期望，但考试计算中通常使用更快的公式 Var(X) = E(X²) − [E(X)]²。

Var(X) = E[(X − μ)²] = Σ (x − μ)² · P(X = x)

Var(X) = E(X²) − [E(X)]²

The standard deviation is the positive square root of the variance and has the same units as X.

标准差是方差的正平方根，并且与 X 的单位相同。

SD(X) = √Var(X)

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