📚 Probability Density Functions | 概率密度函数
In A-Level Statistics, we move beyond discrete random variables to continuous ones where outcomes can take any value in an interval. For these variables, the probability density function (pdf) is the key tool for calculating probabilities. Understanding pdfs is essential for tackling real-world problems involving measurement, time, or any quantity that is not countable.
在A-Level统计学中，我们从离散随机变量扩展到可以在一个区间内取任意值的连续随机变量。对于这些变量，概率密度函数（pdf）是计算概率的关键工具。理解pdf对于处理涉及测量、时间或任何不可计数量的实际问题至关重要。
1. What is a Probability Density Function? | 什么是概率密度函数？
nlike discrete probability mass functions, f(x) itself is not a probability. Instead, the probability that X lies in an interval [a, b] is given by the area under the curve of f(x) over that interval. The pdf must satisfy two conditions: f(x) ≥ 0 for all x, and the total area under the curve over the entire domain equals 1.
概率密度函数通常记作 f(x)，描述了连续随机变量取某个给定值的相对可能性。与离散概率质量函数不同，f(x) 本身不是概率。相反，X 落在区间 [a, b] 内的概率由 f(x) 在该区间上的曲线下方面积给出。pdf 必须满足两个条件：对所有 x 有 f(x) ≥ 0，且整个定义域上曲线下的总面积等于 1。
2. Validating a PDF | 验证一个概率密度函数
r example, if f(x) = kx² for 0 ≤ x ≤ 3 and 0 otherwise, you would set ∫₀³ kx² dx = 1. Solving this gives k = 1/9. Always verify that f(x) ≥ 0 for the given domain.
要检验一个给定的函数是否可作为 pdf，你需要确认其处处非负且整个范围内的积分等于 1。例如，若 f(x) = kx²，0 ≤ x ≤ 3，否则为 0，你会设 ∫₀³ kx² dx = 1。解出 k = 1/9。始终要验证在给定定义域上 f(x) ≥ 0。
3. Calculating Probabilities | 计算概率
, P(a ≤ X ≤ b) = P(a < X < b). Always integrate the pdf over the intersection of the interval [a, b] with the domain where f(x) is non-zero.
对于连续随机变量 X，概率 P(a < X < b) 通过计算 ∫ₐᵇ f(x) dx 得到。由于任何单点的概率为零，所以 P(a ≤ X ≤ b) = P(a < X < b)。始终要在区间 [a, b] 与 f(x) 非零区域的交集上对 pdf 进行积分。
4. The Cumulative Distribution Function (CDF) | 累积分布函数（CDF）
es the probability that X is less than or equal to x. It is defined as F(x) = P(X ≤ x) = ∫-∞x f(t) dt. The pdf is the derivative of the CDF: f(x) = F' ists. Knowing the relationship between pdf and CDF helps when going from one to the other.
累积分布函数 F(x) 给出了 X 小于或等于 x 的概率。它定义为 F(x) = P(X ≤ x) = ∫-∞x f(t) dt。pdf 是 CDF 的导数：在导数存在处有 f(x) = F'(x)。了解 pdf 与 CDF 之间的关系有助于在两者之间转换。
5. Finding the Median from a PDF | 从 PDF 求中位数
) = 0.5, or equivalently P(X ≤ m) = 0.5. To find the median, solve the integral equation ∫-∞m cewise, you may need to determine which section contains the median by evaluating cumulative probabilities step by step.
连续分布的中位数 m 满足 F(m) = 0.5，即 P(X ≤ m) = 0.5。要求中位数，需解积分方程 ∫-∞m f(x) dx = 0.5。如果 pdf 是分段定义的，你可能需要通过逐步评估累积概率来确定中位数位于哪一段。
6. Mode of a Continuous Distribution | 连续分布的众数
alue of x where the pdf f(x) reaches its maximum. To find it, differentiate f(x) and set f' ivative is negative or that it is a genuine maximum within the domain. Some distributions may have more than one mode or no mode if the pdf is constant.
众数是使 pdf f(x) 达到最大的 x 值。要找到它，可对 f(x) 求导并令 f'(x) = 0，然后检查二阶导数为负或确认它在定义域内为真正的最大值。若 pdf 为常数，分布可能没有众数，也可能有多个众数。
7. Expectation and Variance | 期望与方差
riable X is given by E(X) = ∫ x f(x) dx over the entire domain. Variance is Var(X) = E(X²) – ecewise. The formulas are analogous to the discrete case, with sums replaced by integrals.
连续随机变量 X 的期望值（均值）由 E(X) = ∫ x f(x) dx 在整个定义域上给出。方差为 Var(X) = E(X²) – [E(X)]²，其中 E(X²) = ∫ x² f(x) dx。这些积分必须仔细计算，尤其是当 pdf 分段时。这些公式与离散情形类似，只是用积分代替了求和。
8. Percentiles and Quartiles | 百分位数与四分位数
Generalising the median, the 100p-th percentile is the value xp such that F(xp rtile Q1 corresponds to p = 0.25, upper quartile Q3 to p = 0.75. Interquartile range is Q3 – Q1. To find any percentile, solve the integral equation defining F(xp) = p within the appropriate support interval.
将中位数推广，第 100p 百分位数是满足 F(xp) = p 的值 xp。下四分位数 Q1 对应 p = 0.25，上四分位数 Q3 对应 p = 0.75。四分位距为 Q3 – Q1。要求任何百分位数，需在适当的支持区间内解定义 F(xp) = p 的积分方程。
9. Working with Piecewise PDFs | 处理分段概率密度函数
y exam questions define a pdf in pieces. For instance, f(x) = ax for 0 ≤ x < 2, f(x) = b(4 – if the pdf is continuous. Then calculate probabilities or other quantities piece by piece.
许多考题会分段定义 pdf。例如，f(x) = ax，0 ≤ x < 2；f(x) = b(4 – x)，2 ≤ x ≤ 4；在其他地方为 0。你首先利用总面积条件，如果 pdf 连续则用边界处的连续性，来确定未知常数。然后逐段计算概率或其他量。
10. Transforming Variables (Linear) | 变量的线性变换
If Y = aX + b, where X has pdf fX(x), then Y has a pdf given by fY(y) = (1/|a|) fX((y – ear transformations, the pdf method requires using the derivative of the inverse function.
若 Y = aX + b，其中 X 具有 pdf fX(x)，则 Y 的 pdf 由 fY(y) = (1/|a|) fX((y – b)/a) 给出。Y 的取值范围也相应平移。期望和方差遵循规则：E(Y) = aE(X) + b，Var(Y) = a² Var(X)。对于非线性变换，pdf 法需要使用反函数的导数。
11. Practical Applications | 实际应用
onential distribution is often used for the time until an event occurs, and its pdf is f(x) = λe-λx obabilities, means, and percentiles from a pdf helps in decision-making under uncertainty.
概率密度函数可对实际数据建模，如等待时间、身高、测量误差或部件寿命。例如，指数分布常用于事件发生所需的时间，其 pdf 对 x ≥ 0 为 f(x) = λe-λx。能够从 pdf 中提取概率、均值和百分位数有助于在不确定情况下做出决策。
12. Common Mistakes and Exam Tips | 常见错误与考试建议
Students often forget to check that the total area equals 1 or misapply integration limits. Always sketch the pdf if possible to visualize the region for integration. Remember that probabilities for continuous variables are areas, not heights. When finding the median or other percentiles, ensure the cumulative probability equation is set up correctly, and when using a calculator, be precise with your integral bounds.
学生经常忘记检查总面积是否等于 1，或者误用积分限。如果可能的话，始终画出 pdf 的草图以可视化积分区域。记住连续变量的概率是面积，而非高度。求中位数或其他百分位数时，确保累积概率方程设置正确，并在使用计算器时精确给出积分界限。
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