Probability Density Functions and Continuous Distributions | 概率密度函数与连续分布

📚 Probability Density Functions and Continuous Distributions | 概率密度函数与连续分布

In A-level Mathematics, the study of probability extends from discrete random variables to continuous random variables. In the continuous case, probabilities are not assigned to individual points but rather to intervals, and the key tool is the probability density function (PDF).

在 A-level 数学中，概率的研究从离散随机变量延伸到连续随机变量。在连续情形下，概率不再分配给单个点，而是分配给区间，其核心工具是概率密度函数（PDF）。

1. From Discrete to Continuous | 从离散到连续

A discrete random variable takes countable values, and its probability distribution can be represented by a table or a formula P(X = x). For example, rolling a fair die gives each outcome a probability of 1/6.

离散随机变量取可数的值，其概率分布可以用表格或公式 P(X = x) 表示。例如，掷一枚均匀骰子，每个结果出现的概率都是 1/6。

A continuous random variable, however, can take any value within an interval, such as height, weight, time, or temperature. Since there are infinitely many possible values, the probability of any single exact value is effectively zero.

然而，连续随机变量可以在一个区间内取任意值，例如身高、体重、时间或温度。由于可能取值有无限多个，任何一个精确值的概率实际上都是零。

Instead of assigning probabilities to individual points, we describe the distribution using a density curve. The total area under this curve equals 1, and the probability of an interval is the area under the curve over that interval.

我们不再为单个点分配概率，而是用密度曲线来描述分布。这条曲线下方的总面积为 1，而某个区间上的概率就是曲线在该区间下的面积。

2. Defining the Probability Density Function | 概率密度函数的定义

A continuous random variable X has a probability density function f(x) if f(x) satisfies two conditions for all real x:

连续随机变量 X 具有概率密度函数 f(x)，当且仅当 f(x) 对所有实数 x 满足两个条件：

f(x) ≥ 0 for all x. | 对所有 x，f(x) ≥ 0。
The total area under the curve is 1: ∫ f(x) dx = 1 over the entire range. | 曲线下的总面积为 1：在整个范围内 ∫ f(x) dx = 1。

The probability that X lies between a and b is given by the integral:

X 落在 a 与 b 之间的概率由以下积分给出：

P(a ≤ X ≤ b) = ∫ₐᵇ f(x) dx

This integral represents the area under the graph of f(x) from x = a to x = b.

这个积分表示 f(x) 图像从 x = a 到 x = b 下方的面积。

3. Properties of a Valid PDF | 有效概率密度函数的性质

For a function to serve as a probability density function, it must meet two essential properties. Violating either property means the function is not a valid PDF.

一个函数要作为概率密度函数，必须满足两个基本性质。只要违反其中任意一条，该函数就不是有效的 PDF。

Non-negativity: f(x) ≥ 0 for all x in its domain. | 非负性：在其定义域上 f(x) ≥ 0。
Normalization: The integral of f(x) over the entire sample space equals 1. | 归一化：f(x) 在整个样本空间上的积分等于 1。

Because f(x) must be non-negative, the cumulative distribution function F(x) = P(X ≤ x) = ∫₋∞ˣ f(t) dt is always non-decreasing.

由于 f(x) 必须非负，累积分布函数 F(x) = P(X ≤ x) = ∫₋∞ˣ f(t) dt 总是单调不减的。

4. Example: Finding a Constant | 示例：求常数

Consider a continuous random variable X with PDF given by f(x) = kx² for 0 ≤ x ≤ 2, and f(x) = 0 otherwise. To find the value of k, we use the normalization condition.

设连续随机变量 X 的概率密度函数为 f(x) = kx²（当 0 ≤ x ≤ 2），否则 f(x) = 0。为了求 k 的值，我们使用归一化条件。

∫₀² kx² dx = 1

Evaluating the integral gives k × [x³/3]₀² = k × 8/3 = 1, so k = 3/8.

计算积分得 k × [x³/3]₀² = k × 8/3 = 1，因此 k = 3/8。

Thus the complete PDF is f(x) = 3x²/8 for 0 ≤ x ≤ 2. Notice that f(2) = 3 × 4 / 8 = 1.5, which exceeds 1; this is allowed because f(x) is a density, not a probability.

因此完整的 PDF 为 f(x) = 3x²/8（当 0 ≤ x ≤ 2）。注意 f(2) = 3 × 4 / 8 = 1.5，超过 1；这是允许的，因为 f(x) 是密度而不是概率。

5. Computing Probabilities from a PDF | 从概率密度函数计算概率

Once the PDF is fully specified, we can calculate the probability that X falls within any given interval. For instance, using the PDF f(x) = 3x²/8 on [0, 2], we can find P(0.5 ≤ X ≤ 1.5).

一旦 PDF 完全确定，我们就可以计算 X 落在任意给定区间内的概率。例如，使用 [0, 2] 上的 PDF f(x) = 3x²/8，我们可以求 P(0.5 ≤ X ≤ 1.5)。

P(0.5 ≤ X ≤ 1.5) = ∫₀.₅¹·⁵ (3x²/8) dx = [x³/8]₀.₅¹·⁵

This equals (1.5³ − 0.5³)/8 = (3.375 − 0.125)/8 = 3.25/8 = 0.40625. So the probability is approximately 0.406.

这等于 (1.5³ − 0.5³)/8 = (3.375 − 0.125)/8 = 3.25/8 = 0.40625。因此概率约为 0.406。

Notice that this probability is simply the area under the curve between 0.5 and 1.5, not the value of f at any specific point.

注意，这个概率就是曲线在 0.5 到 1.5 之间的面积，而不是 f 在某个具体点的取值。

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

The cumulative distribution function (CDF), denoted F(x), gives the probability that X takes a value less than or equal to x:

累积分布函数（CDF），记作 F(x)，表示 X 取值小于或等于 x 的概率：

F(x) = P(X ≤ x) = ∫₋∞ˣ f(t) dt

The CDF is useful because probabilities of intervals can be expressed as differences: P(a ≤ X ≤ b) = F(b) − F(a).

CDF 很有用，因为区间的概率可以表示为差值：P(a ≤ X ≤ b) = F(b) − F(a)。

For the example above, F(x) = 0 for x < 0, F(x) = x³/8 for 0 ≤ x ≤ 2, and F(x) = 1 for x > 2. The CDF is always non-decreasing and approaches 0 as x → −∞ and 1 as x → ∞.

对于上面的例子，当 x < 0 时 F(x) = 0；当 0 ≤ x ≤ 2 时 F(x) = x³/8；当 x > 2 时 F(x) = 1。CDF 总是单调不减的，并且当 x → −∞ 时趋向 0，当 x → ∞ 时趋向 1。

7. The Mean of a Continuous Distribution | 连续分布的均值

The mean, or expected value, of a continuous random variable X with PDF f(x) is defined as:

连续随机变量 X（具有 PDF f(x)）的均值或期望值定义为：

E(X) = ∫ x f(x) dx

The integral is taken over the entire range of X. For our example, E(X) = ∫₀² x × (3x²/8) dx = ∫₀² (3x³/8) dx = 3/8 × [x⁴/4]₀² = 3/8 × 16/4 = 3/8 × 4 = 1.5.

积分在 X 的整个范围内进行。在我们的例子中，E(X) = ∫₀² x × (3x²/8) dx = ∫₀² (3x³/8) dx = 3/8 × [x⁴/4]₀² = 3/8 × 16/4 = 3/8 × 4 = 1.5。

Compare this to the discrete case, where E(X) = Σ x P(X = x). The integral plays the role of summation in the continuous setting.

这与离散情形 E(X) = Σ x P(X = x) 形成对比。在连续情形中，积分扮演了求和的角色。

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

The variance of a continuous random variable is defined as:

连续随机变量的方差定义为：

Var(X) = E(X²) − [E(X)]²

where E(X²) = ∫ x² f(x) dx. The standard deviation is σ = √Var(X).

其中 E(X²) = ∫ x² f(x) dx。标准差为 σ = √Var(X)。

For our example, E(X²) = ∫₀² x² × (3x²/8) dx = ∫₀² (3x⁴/8) dx = 3/8 × [x⁵/5]₀² = 3/8 × 32/5 = 12/5 = 2.4.

对于我们的例子，E(X²) = ∫₀² x² × (3x²/8) dx = ∫₀² (3x⁴/8) dx = 3/8 × [x⁵/5]₀² = 3/8 × 32/5 = 12/5 = 2.4。

Therefore Var(X) = 2.4 − 1.5² = 2.4 − 2.25 = 0.15, and σ = √0.15 ≈ 0.387.

因此 Var(X) = 2.4 − 1.5² = 2.4 − 2.25 = 0.15，σ = √0.15 ≈ 0.387。

9. The Uniform Distribution | 均匀分布

One of the simplest continuous distributions is the uniform distribution on the interval [a, b], denoted U(a, b). Its PDF is constant over the interval:

最简单的连续分布之一是在区间 [a, b] 上的均匀分布，记作 U(a, b)。其 PDF 在区间上为常数：

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

The mean is E(X) = (a + b)/2, the midpoint of the interval. The variance is Var(X) = (b − a)²/12.

均值为 E(X) = (a + b)/2，即区间的中点。方差为 Var(X) = (b − a)²/12。

For example, if X ~ U(0, 10), then P(2 ≤ X ≤ 5) = (5 − 2)/(10 − 0) = 3/10 = 0.3.

例如，若 X ~ U(0, 10)，则 P(2 ≤ X ≤ 5) = (5 − 2)/(10 − 0) = 3/10 = 0.3。

10. The Exponential Distribution | 指数分布

The exponential distribution is often used to model waiting times or lifetimes. Its PDF is:

指数分布常用于建模等待时间或寿命。其 PDF 为：

f(x) = λe^(−λx) for x ≥ 0, 0 otherwise

Here λ > 0 is the rate parameter. The mean is E(X) = 1/λ, and the variance is Var(X) = 1/λ².

这里 λ > 0 是速率参数。均值为 E(X) = 1/λ，方差为 Var(X) = 1/λ²。

The CDF of the exponential distribution has a simple closed form: F(x) = 1 − e^(−λx) for x ≥ 0. This makes it easy to compute probabilities such as P(X > t) = e^(−λt).

指数分布的 CDF 具有简单的封闭形式：当 x ≥ 0 时 F(x) = 1 − e^(−λx)。这使得计算概率非常方便，例如 P(X > t) = e^(−λt)。

11. The Normal Distribution | 正态分布

The normal distribution is the most important continuous distribution in statistics. Its PDF is given by:

正态分布是统计学中最重要的连续分布。其 PDF 为：

f(x) = 1/(σ√(2π)) × e^(−(x−μ)²/(2σ²))

Here μ is the mean and σ is the standard deviation. The curve is symmetric about x = μ, and the total area under the curve is 1.

这里 μ 是均值，σ 是标准差。曲线关于 x = μ 对称，曲线下总面积为 1。

A standard normal distribution is one with μ = 0 and σ = 1. Any normal random variable X ~ N(μ, σ²) can be transformed to Z = (X − μ)/σ, which follows N(0, 1).

标准正态分布是 μ = 0、σ = 1 的正态分布。任何正态随机变量 X ~ N(μ, σ²) 都可以通过 Z = (X − μ)/σ 转换为服从 N(0, 1) 的标准正态变量。

12. Applying Continuous Distributions | 连续分布的应用

When solving problems involving continuous distributions, always verify that the PDF is properly normalized before computing probabilities or expectations. A quick check is to integrate f(x) over the full range and confirm the result is 1.

在解决涉及连续分布的问题时，务必先验证 PDF 是否正确归一化，然后再计算概率或期望。快速检查方法是在整个范围内对 f(x) 积分，确认结果为 1。

Remember that P(X = c) = 0 for any continuous random variable, so statements like P(X < a) and P(X ≤ a) give the same value. This subtle point is a common exam trap.

记住，对于任何连续随机变量，P(X = c) = 0，因此 P(X < a) 与 P(X ≤ a) 的值相同。这个细微之处是常见的考试陷阱。

Finally, always draw a sketch of the density curve when working with intervals. Visualizing the area you are calculating helps prevent errors in limits and signs.

最后，在处理区间问题时，务必画出密度曲线的草图。可视化你正在计算的面积有助于避免在积分限和符号上出错。

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Probability Density Functions and Continuous Distributions | 概率密度函数与连续分布