Probability Density Functions | 概率密度函数

📚 Probability Density Functions | 概率密度函数

In A-Level Mathematics, many real-world quantities such as time, length and temperature are modelled as continuous random variables. A probability density function, or PDF, describes how the total probability is distributed over all possible values of a continuous random variable. Unlike discrete probability mass functions, a PDF does not give a probability at a single point; instead, probabilities are found by integrating the PDF over an interval.

在 A-Level 数学中，时间、长度和温度等许多现实量被建模为连续随机变量。概率密度函数（PDF）描述总概率如何在连续随机变量的所有可能取值上分布。与离散概率质量函数不同，PDF 不给出某一点的概率；概率是通过对区间上的 PDF 进行积分得到的。

1. What Is a Probability Density Function? | 什么是概率密度函数？

A continuous random variable X takes values in an interval rather than isolated points. Its probability density function f(x) is a non-negative function such that the area under the curve y = f(x) between x = a and x = b gives P(a ≤ X ≤ b).

连续随机变量 X 在区间上取值，而不是孤立点。其概率密度函数 f(x) 是一个非负函数，曲线 y = f(x) 在 x = a 与 x = b 之间的面积给出 P(a ≤ X ≤ b)。

The notation f(x) is used because it acts like a density: the total mass is 1, and probability mass is accumulated by area, not by height.

使用记号 f(x) 是因为它像密度一样：总质量为 1，概率质量通过面积而非高度累积。

P(a ≤ X ≤ b) = ∫ₐᵇ f(x) dx

2. Key Conditions for a Valid PDF | 概率密度函数的基本条件

A function f(x) can serve as a probability density function for a continuous random variable only if two conditions hold.

函数 f(x) 要作为连续随机变量的概率密度函数，必须满足两个条件。

First, f(x) must be non-negative everywhere: f(x) ≥ 0 for all x in the domain.

第一，f(x) 必须处处非负：对所有定义域内的 x 都有 f(x) ≥ 0。

Second, the total area under the curve must equal 1, which is called the normalisation condition.

第二，曲线下的总面积必须等于 1，这称为归一化条件。

∫₋∞⁺∞ f(x) dx = 1

If either condition fails, the function cannot be a PDF. These conditions are often used to find unknown constants in exam questions.

如果任一条件不满足，该函数就不能作为 PDF。考试题常利用这些条件求未知常数。

3. Calculating Probabilities from a PDF | 利用概率密度函数求概率

For a continuous random variable X with density f(x), the probability that X lies between a and b is found by definite integration.

对于具有密度 f(x) 的连续随机变量 X，X 落在 a 与 b 之间的概率用定积分求得。

P(a < X < b) = P(a ≤ X ≤ b) = ∫ₐᵇ f(x) dx

Since the probability of any single point is zero for a continuous variable, strict and non-strict inequalities produce the same result.

由于连续变量在任意单点的概率为零，严格不等式与非严格不等式给出相同结果。

Always check that the interval lies within the support where f(x) is defined and integrate only over the overlap.

务必检查区间是否位于 f(x) 定义的支撑集内，并只对重叠部分积分。

4. The Cumulative Distribution Function | 累积分布函数

The cumulative distribution function (CDF) gives the probability that X is at most x.

累积分布函数（CDF）给出 X 不超过 x 的概率。

F(x) = P(X ≤ x) = ∫₋∞ˣ f(t) dt

The CDF is useful because F(b) − F(a) = P(a < X ≤ b). It is non-decreasing and satisfies F(x) → 0 as x → −∞ and F(x) → 1 as x → ∞.

CDF 很有用，因为 F(b) − F(a) = P(a < X ≤ b)。它是非递减的，且满足当 x → −∞ 时 F(x) → 0，当 x → ∞ 时 F(x) → 1。

If a question provides F(x), the PDF can be recovered by differentiating: f(x) = F′(x), wherever F is differentiable.

如果题目给出 F(x)，可通过求导还原 PDF：在 F 可导处 f(x) = F′(x)。

5. Finding the Median and Quartiles | 求中位数与四分位数

The median m of a continuous distribution is the value such that half the total probability lies below it.

连续分布的中位数 m 是使总概率的一半位于其下方的值。

∫₋∞ᵐ f(x) dx = 0.5

The lower quartile Q₁ satisfies F(Q₁) = 0.25, and the upper quartile Q₃ satisfies F(Q₃) = 0.75. The interquartile range is Q₃ − Q₁.

下四分位数 Q₁ 满足 F(Q₁) = 0.25，上四分位数 Q₃ 满足 F(Q₃) = 0.75。四分位距为 Q₃ − Q₁。

Solve these equations using the CDF and remember to check that the solution lies in the support of X.

利用 CDF 解这些方程，并记得检查解是否在 X 的支撑集内。

6. Mode of a Continuous Distribution | 连续分布的众数

The mode of a continuous distribution is the value of x that maximises the PDF f(x), provided the PDF has a maximum within the support.

连续分布的众数是使 PDF f(x) 最大的 x 值，前提是 PDF 在支撑集内存在最大值。

To find it, differentiate f(x), set f′(x) = 0, and check the second derivative or use a sign test. If f(x) has no turning point, the mode may occur at an endpoint.

求众数时，对 f(x) 求导，令 f′(x) = 0，并检查二阶导数或使用符号检验。如果 f(x) 没有驻点，众数可能出现在端点。

Note that the mode is not found from F(x) directly; it is a property of the density curve.

注意众数不能直接从 F(x) 求得；它是密度曲线本身的属性。

7. Expectation and Variance | 期望与方差

The expectation or mean of X is the centre of mass of the distribution.

X 的期望或均值是分布的质量中心。

E(X) = ∫₋∞⁺∞ x f(x) dx

The variance is defined as Var(X) = E(X²) − [E(X)]², where E(X²) is the second moment.

方差定义为 Var(X) = E(X²) − [E(X)]²，其中 E(X²) 是二阶矩。

E(X²) = ∫₋∞⁺∞ x² f(x) dx

These integrals may be improper; compute them over the support only. The standard deviation is the positive square root of the variance.

这些积分可能是广义积分；只在支撑集上计算。标准差是方差的正平方根。

8. Common Continuous Distributions | 常见连续分布

Several named distributions appear frequently at A-Level.

几个常见分布在 A

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