Continuous Distributions | 连续分布

📚 Continuous Distributions | 连续分布

In A-Level Mathematics, many random variables can take any value within an interval. These are called continuous random variables. Understanding continuous distributions is essential for modelling measurements such as time, height, mass and temperature. This article covers probability density functions, cumulative distribution functions, expectation, variance, percentiles, the uniform distribution and the normal distribution.

在A-Level数学中，许多随机变量可以取某一区间内的任意值，这类变量称为连续随机变量。理解连续分布对于建立时间、身高、质量和温度等测量数据的模型至关重要。本文涵盖概率密度函数、累积分布函数、期望、方差、百分位数、均匀分布以及正态分布。

1. Continuous Random Variables | 连续随机变量

A continuous random variable can take any value in a given interval, so the number of possible values is uncountably infinite. Unlike discrete variables, we cannot list every possible outcome.

连续随机变量可以取给定区间内的任意值，因此其可能取值的数量是不可数无限的。与离散变量不同，我们无法列出每一个可能的结果。

Probabilities are assigned to intervals, such as P(a < X < b), rather than to individual points. For any continuous random variable X, the probability of any single value is zero, so P(X = x) = 0 for every x.

概率被分配给区间，例如 P(a < X < b)，而不是分配给单个点。对于任何连续随机变量 X，任何单点值的概率都为零，因此对每个 x 都有 P(X = x) = 0。

2. Probability Density Function (PDF) | 概率密度函数

A probability density function, usually written f(x), describes the relative likelihood of a continuous random variable taking a value near x. The height f(x) is not itself a probability, but the area under the curve y = f(x) gives probability.

概率密度函数通常写作 f(x)，描述连续随机变量在 x 附近取值的相对可能性。高度 f(x) 本身不是概率，但曲线 y = f(x) 下方的面积给出概率。

Probabilities are found by integrating the pdf over an interval:

概率通过对概率密度函数在区间上积分求得：

P(a < X < b) = ∫ₐᵇ f(x) dx

The area under the curve y = f(x) between x = a and x = b represents the probability P(a < X < b). The total area under a valid pdf over its entire domain equals 1.

曲线 y = f(x) 下介于 x = a 和 x = b 之间的面积表示概率 P(a < X < b)。有效概率密度函数在其整个定义域上的总面积等于 1。

3. Valid PDF Conditions | 有效概率密度函数的条件

For f(x) to be a valid probability density function, two conditions must hold:

要使 f(x) 成为有效的概率密度函数，必须满足两个条件：

f(x) ≥ 0 for all x in the domain | 对所有定义域内的 x，f(x) ≥ 0
∫ f(x) dx = 1 over the whole domain | 在整个定义域上积分为 1

If a function is given as a piecewise definition, check that each piece is non-negative and that the total integral is 1. A common exam task is to find an unknown constant k by setting the total integral equal to 1.

如果函数以分段形式给出，需要检查每一段都为非负，并且总积分等于1。常见的考试题型是令总积分等于1，求出未知常数 k。

4. Cumulative Distribution Function (CDF) | 累积分布函数

The cumulative distribution function F(x) gives the probability that X is less than or equal to x:

累积分布函数 F(x) 给出 X 小于或等于 x 的概率：

F(x) = P(X ≤ x) = ∫ from −∞ to x f(t) dt

For continuous distributions, F(x) is a continuous, non-decreasing function with F(x) → 0 as x → −∞ and F(x) → 1 as x → ∞.

对于连续分布，F(x) 是一个连续且非递减的函数，当 x → −∞ 时 F(x) → 0，当 x → ∞ 时 F(x) → 1。

If the pdf is defined in pieces, the CDF is also found piecewise by integrating each section carefully and ensuring the values match at the interval boundaries.

如果概率密度函数分段定义，累积分布函数也通过逐段积分得到，并确保各段在区间边界处的取值吻合。

5. Connecting PDF and CDF | 概率密度函数与累积分布函数的关系

The pdf is the derivative of the CDF: f(x) = F′(x). Conversely, the CDF is the integral of the pdf from the lower limit of the distribution up to x.

概率密度函数是累积分布函数的导数：f(x) = F′(x)。反过来，累积分布函数是概率密度函数从分布下限到 x 的积分。

This relationship allows you to move between f(x) and F(x) in both directions, which is a common A-Level skill. When F(x) is given and you need a probability, simply evaluate F(b) − F(a).

这种关系允许你在 f(x) 和 F(x) 之间双向转换，这是A-Level常见技能。当给出 F(x) 并需要概率时，只需计算 F(b) − F(a)。

P(a < X < b) = F(b) − F(a)

6. Mean and Variance | 均值与方差

The mean or expected value of a continuous random variable X is given by E(X) = ∫ x f(x) dx over the whole domain.

连续随机变量 X 的均值或期望为 E(X) = ∫ x f(x) dx，在整个定义域上积分。

The variance is Var(X) = E(X²) − [E(X)]², where E(X²) = ∫ x² f(x) dx. The standard deviation is σ = √Var(X).

方差为 Var(X) = E(X²) − [E(X)]²，其中 E(X²) = ∫ x² f(x) dx。标准差为 σ = √Var(X)。

E(X) = ∫ x f(x) dx, Var(X) = ∫ x² f(x) dx − μ²

These formulas are similar to those for discrete variables, but sums are replaced by integrals.

这些公式与离散随机变量的公式类似，只是求和被积分替代。

7. Median and Percentiles | 中位数与百分位数

The median m satisfies F(m) = 0.5, or equivalently P(X ≤ m) = 0.5. To find the lower quartile Q₁, solve F(Q₁) = 0.25, and for the upper quartile Q₃, solve F(Q₃) = 0.75.

中位数 m 满足 F(m) = 0.5，等价于 P(X ≤ m) = 0.5。求下四分位数 Q₁ 时解 F(Q₁) = 0.25，求上四分位数 Q₃ 时解 F(Q₃) = 0.75。

The pth percentile is found by solving F(x) = p/100, where p is between 0 and 100. If only the pdf is given, integrate it from the lower limit up to an unknown value and set the result equal to the required probability.

第 p 百分位数通过解 F(x) = p/100 得到，其中 p 在 0 到 100 之间。如果只给出概率密度函数，则从下限积分到某个未知值，令结果等于所需概率。

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

The mode of a continuous distribution is the value of x where the pdf f(x) reaches its maximum. To find the mode, differentiate f(x), set f′(x) = 0, and check that the second derivative is negative or that the sign of f′(x) changes from positive to negative.

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Continuous Distributions | 连续分布