IB Mathematics: Mean and Variance of Continuous Random Variables | IB数学：连续随机变量的均值与方差

📚 IB Mathematics: Mean and Variance of Continuous Random Variables | IB数学：连续随机变量的均值与方差

In this article we explore how to calculate the mean and variance of a continuous random variable, a key topic in IB Mathematics Higher Level. We will review the definition of a probability density function, interpret the mean as a long-run average, and derive the variance using the shortcut formula.

本文将探讨连续随机变量均值与方差的计算方法，这是IB数学高级水平中的一个重点课题。我们将回顾概率密度函数的定义，把均值理解为长期平均值，并通过捷径公式推导方差。

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1. Probability Density Functions | 概率密度函数

A continuous random variable X takes values in an interval or a union of intervals. Its probability distribution is described by a probability density function (pdf), written f(x). Probabilities are found by integrating the pdf over the required interval.

连续随机变量X在一个区间或若干个区间的并集上取值。其概率分布由概率密度函数（pdf）f(x)描述。通过将pdf在所需区间上积分来求概率。

The probability that X lies between a and b is the area under the curve from a to b:

X落在a与b之间的概率是从a到b的曲线下面积：

P(a ≤ X ≤ b) = ∫_a^b f(x) dx

Because X can take infinitely many values, the probability of any single exact value is zero. We only work with intervals of values.

因为X可能取值无穷多个，任何单一具体值的概率都为零。我们只处理取值区间。

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2. Conditions for a Valid PDF | 概率密度函数的有效性条件

A function f(x) can serve as a pdf only if two conditions hold. First, f(x) must be non-negative for every x in the sample space:

函数f(x)要成为概率密度函数，必须满足两个条件。第一，在样本空间中每个x处有 f(x) 非负：

f(x) ≥ 0 对所有x

Second, the total area under the graph of f(x) must equal 1, because the total probability must be 1:

第二，f(x)曲线下的总面积必须等于1，因为总概率必须为1：

∫_-∞^+∞ f(x) dx = 1

If a pdf is defined only on a finite interval [A, B], the integral is taken from A to B instead of −∞ to +∞.

如果pdf只在有限区间[A, B]上有定义，则积分取A到B而非−∞到+∞。

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3. The Mean of a Continuous Random Variable | 连续随机变量的均值

The mean, or expected value, of a continuous random variable X is denoted by E(X). It is defined as the weighted average of all possible values, with weights given by the pdf.

连续随机变量X的均值或期望值记为E(X)。它定义为所有可能取值的加权平均值，权重由pdf提供。

E(X) = ∫_-∞^+∞ x f(x) dx

If the pdf is zero outside an interval [A, B], we may integrate from A to B only.

如果pdf在区间[A, B]之外为零，则只需从A积分到B。

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4. Interpretation of the Mean | 均值的含义

The mean is the centre of mass of the probability distribution. If the distribution is symmetric, the mean equals the axis of symmetry. The mean may not be a possible value of X; it is the long-run average of many observations.

均值是概率分布的质心。若分布对称，均值等于对称轴。均值不一定是X的可能取值；它是大量观测的长期平均值。

For a discrete random variable the expected value is a sum, but for a continuous random variable it is an integral. The integral can often be evaluated by inspection when the pdf is symmetric.

对于离散随机变量，期望值是求和；对于连续随机变量，期望值是积分。当pdf对称时，常可直接看出积分值。

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

The variance of X measures the expected squared deviation from the mean. Let μ = E(X). Then the variance is defined as:

X的方差度量的是与均值的期望平方偏差。设 μ = E(X)，则方差定义为：

Var(X) = 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)

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6. The Shortcut Formula for Variance | 方差的捷径公式

To avoid integrating a squared binomial, we use the shortcut formula. It states that the variance equals the mean of the squares minus the square of the mean:

为避免对二项式平方积分，我们使用捷径公式。方差等于平方的均值减去均值的平方：

Var(X) = E(X²) − [E(X)]²

Here E(X²) is the second moment, computed as:

其中E(X²)是二阶矩，计算方式为：

E(X²) = ∫_-∞^+∞ x² f(x) dx

This formula is usually simpler in examinations because it avoids expanding (x − μ)².

这个公式在考试中通常更简单，因为它避免了展开(x − μ)²。

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7. Example: Continuous Uniform Distribution | 例：连续均匀分布

A random variable X is uniformly distributed on [a, b] if its pdf is constant on that interval:

如果随机变量X的概率密度函数在区间[a, b]上为常数，则称X服从[a, b]上的均匀分布：

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

The mean is the midpoint of the interval:

均值是区间的中点：

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

The variance is:

方差为：

Var(X) = (b − a)²/12

These results are standard and can be derived by direct integration of the pdf.

这些结果是标准的，可以通过对pdf直接积分得到。

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8. Example: Exponential Distribution | 例：指数分布

An exponential random variable with rate λ > 0 has pdf:

参数为λ > 0的指数随机变量的pdf为：

f(x) = λe^−λx for x ≥ 0

It is commonly used to model waiting times. Its mean and variance are:

指数分布常用于建模等待时间。其均值与方差为：

E(X) = 1/λ, Var(X) = 1/λ²

Note that some textbooks write the pdf as (1/μ)e^−x/μ, where μ = 1/λ. Always check which parameterisation your IB formula booklet uses.

注意有些教材把pdf写成(1/μ)e^−x/μ，其中μ = 1/λ。务必查看你的IB公式手册使用哪种参数化。

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9. Linear Transformations | 线性变换

If X has mean μ and variance σ², then the random variable Y = aX + b (with constants a and b) has:

若X的均值为μ，方差为σ²，则随机变量Y = aX + b（其中a和b为常数）有：

E(Y) = aμ + b, Var(Y) = a²σ²

This result is very useful for standardising normal variables. If Z = (X − μ)/σ, then E(Z) = 0 and Var(Z) = 1.

这个结果对标准化正态变量非常有用。若 Z = (X − μ)/σ，则 E(Z) = 0，Var(Z) = 1。

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10. Common Errors and Exam Tips | 常见错误与考试技巧

• Forgetting to check that the pdf integrates to 1 before using it. This is a common source of lost marks.

  使用pdf前忘记检查它积分为1。这是常见的失分点。

• Confusing f(x) with a probability. For a continuous random variable, f(x) is a density, not a probability.

  把f(x)误认为概率。对于连续随机变量，f(x)是密度，而不是概率。

• Using the variance shortcut formula but forgetting to subtract [E(X)]². Always compute both E(X²) and E(X) separately.

  使用方差捷径公式时忘记减去[E(X)]²。务必分别计算E(X²)和E(X)。

• Using incorrect limits when the distribution has finite support. If f(x) = 0 outside [A, B], integrate from A to B.

  当分布有有限定义域时使用错误的积分上下限。若f(x)在[A, B]之外为零，则从A积分到B。

• Not dropping a constant correctly under a linear transformation: Var(aX + b) = a²Var(X), so the constant b disappears.

  在线性变换中没有正确处理常数：Var(aX + b) = a²Var(X)，所以常数b消失。

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