Mean and variance of a continuous distribution | 连续分布的均值与方差

📚 Mean and variance of a continuous distribution | 连续分布的均值与方差

In A-Level Mathematics, continuous random variables are described by a probability density function (PDF). This article explains how to find the mean (expectation) and variance of a continuous distribution, including key formulas, common distributions and exam-style techniques.

在A-Level数学中，连续随机变量由概率密度函数（PDF）描述。本文讲解如何求连续分布的均值（期望）和方差，包括关键公式、常见分布和考试题技巧。

1. Continuous random variables and PDFs | 连续随机变量与概率密度函数

A continuous random variable X can take any value in an interval. Its probability distribution is defined by a probability density function f(x), which must satisfy two conditions:

连续随机变量X可以取某个区间内的任意值。其概率分布由概率密度函数f(x)定义，该函数必须满足两个条件：

f(x) ≥ 0 for all x

对所有x，f(x) ≥ 0
∫_-∞^∞ f(x) dx = 1

∫_-∞^∞ f(x) dx = 1

The probability that X lies between two values a and b is the area under the curve: P(a < X < b) = ∫_a^b f(x) dx.

X落在两个值a和b之间的概率是曲线下的面积：P(a < X < b) = ∫_a^b f(x) dx。

In Edexcel exam questions, you are often given a function f(x) on a specific interval such as [0, 2] or [1, 4], and f(x) = 0 elsewhere. The interval where f(x) > 0 is called the support of the distribution.

在Edexcel考试题中，你经常会被给出在特定区间（如[0, 2]或[1, 4]）上的函数f(x)，其他地方f(x) = 0。f(x) > 0的区间称为该分布的支持区间。

2. The mean (expectation) of a continuous distribution | 连续分布的均值（期望）

The mean or expected value of a continuous random variable X is denoted E(X) or μ. It is defined by the integral:

连续随机变量X的均值或期望值记为E(X)或μ。其定义为积分：

E(X) = μ = ∫_-∞^∞ x f(x) dx

If f(x) is zero outside an interval [a, b], the integral is evaluated from a to b. The mean represents the long-run average value of X.

如果f(x)在区间[a, b]之外为零，则积分从a到b计算。均值代表X的长期平均值。

Do not confuse the mean with the median or mode. The mean can be pulled away by a long tail of the distribution, so in a skewed distribution the mean is often closer to the tail than the median is.

不要把均值与中位数或众数混淆。均值可能会被分布的长尾拉偏，因此在偏态分布中，均值通常比中位数更靠近尾部。

3. The variance and standard deviation | 方差与标准差

The variance measures the spread of the distribution around the mean. It is denoted Var(X) or σ² and is defined by:

方差衡量分布围绕均值的离散程度。它记为Var(X)或σ²，定义为：

Var(X) = σ² = E[(X – μ)²] = ∫_-∞^∞ (x – μ)² f(x) dx

The standard deviation is the positive square root of the variance:

标准差是方差的正平方根：

σ = √Var(X)

A larger variance means the values of X are more spread out. A smaller variance means they are more concentrated around the mean.

方差越大意味着X的取值越分散。方差越小意味着取值更集中在均值附近。

In the Edexcel specification, you may be asked to calculate Var(X) directly from its definition or by using the alternative formula, which is usually quicker.

在Edexcel考试大纲中，你可能会被要求直接根据定义计算Var(X)，或使用通常更快捷的替代公式。

4. Alternative variance formula | 方差的计算公式

In practice, it is often easier to use the alternative formula for variance:

在实际计算中，通常使用方差的另一种公式更为简便：

Var(X) = E(X²) – [E(X)]² = ∫_-∞^∞ x² f(x) dx – μ²

This formula avoids subtracting the mean inside the square. You first find E(X), then E(X²), and substitute.

该公式避免了在平方内减去均值。你先求E(X)，再求E(X²)，然后代入。

Be careful: E(X²) is not the same as [E(X)]². Many exam errors come from mixing these two quantities.

注意：E(X²)与[E(X)]²并不相同。许多考试错误都源于混淆这两个量。

This formula also shows that Var(X) is always non-negative, because it can be written as E(X²) – μ² where E(X²) ≥ μ² by a well-known result for random variables.

该公式还表明Var(X)始终为非负值，因为它可写成E(X²) – μ²，而对于随机变量有E(X²) ≥ μ²这一已知结论。

5. Linear transformations: E(aX+b) and Var(aX+b) | 线性变换：E(aX+b) 与 Var(aX+b)

If a and b are constants, then the expectation and variance of the transformed variable Y = aX + b are given by:

如果a和b是常数，则变换后的变量Y = aX + b的期望和方差由下式给出：

E(aX + b) = aE(X) + b

Var(aX + b) = a² Var(X)

Adding b shifts the mean but does not change the variance. Multiplying by a scales the mean by a and the variance by a².

加上b会使均值平移，但不改变方差。乘以a会使均值变为a倍，方差变为a²倍。

The standard deviation of aX + b is |a|σ, because σ is always non-negative. This means if a = -2 and b = 3, then Var(-2X + 3) = 4 Var(X) and the standard deviation is 2σ.

aX + b的标准差为|a|σ，因为σ始终为非负值。这意味着如果a = -2且b = 3，则Var(-2X + 3) = 4 Var(X)，标准差为2σ。

6. The continuous uniform distribution | 连续均匀分布

The continuous uniform distribution over [a, b] has constant density:

在[a, b]上的连续均匀分布具有恒定的密度：

f(x) = 1/(b – a) for a ≤ x ≤ b, and 0 otherwise

Its mean and variance are:

其均值和方差为：

E(X) = (a + b)/2

Var(X) = (b – a)²/12

For example, if X ~ U(0, 4), then E(X) = 2 and Var(X) = 16/12 = 4/3.

例如，如果X ~ U(0, 4)，则E(X) = 2，Var(X) = 16/12 = 4/3。

In exam questions, you may need to derive these results by integrating x f(x) and x² f(x) rather than quoting them, so always show the integration unless the question says otherwise.

在考试题中，你可能需要通过积分x f(x)和x² f(x)来推导这些结果，而不是直接引用，因此除非题目另有说明，始终展示积分过程。

7. The normal distribution: mean and variance parameters | 正态分布：均值与方差参数

If X follows a normal distribution with mean μ and variance σ², we write X ~ N(μ, σ²). The PDF is:

如果X服从均值μ、方差σ²的正态分布，我们记为X ~ N(μ, σ²)。其概率密度函数为：

f(x) = (1/(σ√(2π))) × e^[-(x – μ)²/(2σ²)]

The mean and variance are parameters of the distribution, so no integration is needed. The standard normal distribution has μ = 0 and σ² = 1.

均值和方差是分布的参数，因此无需积分。标准正态分布的μ = 0，σ² = 1。

In Edexcel questions, you often use the normal distribution to model real-world data and calculate probabilities with the standard normal variable Z = (X – μ)/σ.

在Edexcel考试题中，你经常使用正态分布对实际数据建模，并通过标准正态变量Z = (X – μ)/σ计算概率。

Even though the normal PDF is symmetric, the majority of the probability lies within three standard deviations of the mean. This is often tested when finding interval probabilities.

尽管正态PDF是对称的，但大部分概率位于均值三个标准差之内。这一点经常在求区间概率时被考查。

8. Cumulative distribution function and median/percentiles | 累积分布函数与中位数、百分位数

The cumulative distribution function (CDF) gives P(X ≤ x). It is obtained by integrating the PDF:

累积分布函数（CDF）给出P(X ≤ x)。它通过对PDF积分得到：

F(x) = P(X ≤ x) = ∫_-∞^x f(t) dt

The median m of a continuous distribution satisfies F(m) = 0.5. Other percentiles are found by solving F(p) = k/100.

连续分布的中位数m满足F(m) = 0.5。其他百分位数通过解F(p) = k/100求得。

The mean, median and mode coincide only for symmetric unimodal distributions such as the normal distribution. In a skewed distribution, they differ.

均值、中位数和众数仅在对称单峰分布（如正态分布）中重合。在偏态分布中，它们各不相同。

Be careful with the lower limit of the CDF: if the support starts at a, then F(x) = ∫_a^x f(t) dt for x within the support, and F(x) = 0 for x below a, and F(x) = 1 for x above b.

注意CDF的下限：如果支持区间从a开始，则对于支持区间内的x，F(x) = ∫_a^x f(t) dt；当x低于a时F(x) = 0，当x高于b时F(x) = 1。

9. Worked example: finding mean and variance from a given pdf | 例题：从给定概率密度函数求均值与方差

Question: A continuous random variable X has PDF f(x) = 3x² for 0 ≤ x ≤ 1, and 0 otherwise. Find E(X), Var(X) and σ.

问题：连续随机变量X的概率密度函数为f(x) = 3x²（0 ≤ x ≤ 1），其他为0。求E(X)、Var(X)和σ。

Step 1: E(X) = ∫₀¹ x × 3x

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Mean and variance of a continuous distribution | 连续分布的均值与方差