📚 Probability Density Functions | 概率密度函数
A probability density function (PDF) is a fundamental concept in continuous probability distributions, used extensively in Edexcel A Level Mathematics. It describes the relative likelihood of a continuous random variable taking on a given value, with the total area under its curve equal to 1. Understanding PDFs is essential for tackling problems involving cumulative distribution functions, medians, modes, and expectations.
概率密度函数是连续型概率分布的核心概念，在Edexcel A Level数学中应用广泛。它描述了连续随机变量取某个特定值的相对可能性，其曲线下的总面积等于1。理解概率密度函数对解决累积分布函数、中位数、众数和期望等问题至关重要。
1. Definition of a Probability Density Function | 概率密度函数的定义
domain, and the total area under the curve equals 1, i.e., ∫ f(x) dx over the whole domain = 1.
若函数 f(x) 是连续随机变量 X 的概率密度函数，必须满足两个关键条件：在其定义域内所有 x 都有 f(x) ≥ 0，且曲线下的总面积等于1，即 ∫ f(x) dx（在整个定义域上）= 1。
The probability that X lies in the interval [a, b] is given by the definite integral ∫ab . For a continuous variable, the probability at any single point is zero, so P(X = a) = 0.
X 落在区间 [a, b] 内的概率由定积分 ∫ab f(x) dx 给出。对于连续变量，单点的概率为零，故 P(X = a) = 0。
2. Verifying a Valid PDF | 验证有效的概率密度函数
To verify that a given function is a PDF, we must check non‑negativity and integrate over the entire range to confirm the area is 1. For piecewise functions, integration is performed over each piece.
要验证一个给定函数是概率密度函数，我们必须检查其非负性，并在整个范围内积分以确认面积为1。对于分段函数，需分别对每一段进行积分。
3/4. Then check f(x) ≥ 0 on [0,2]: 3/4 x(2 − x) ≥ 0 since x(2 − x) ≥ 0 for 0 ≤ x ≤ 2. Hence it is a PDF.
例：f(x) = kx(2 − x)，0 ≤ x ≤ 2，其余为0。首先求 k 使面积为1。对 kx(2 − x) 从0到2积分得 k(4 − 8/3) = (4/3)k = 1，故 k = 3/4。然后检查 f(x) ≥ 0 在 [0,2] 上：3/4 x(2 − x) ≥ 0，因为当 0 ≤ x ≤ 2 时 x(2 − x) ≥ 0。因此它是一个概率密度函数。
3. Cumulative Distribution Function (CDF) | 累积分布函数
The cumulative distribution function F(x) = P(X ≤ x) = ∫−∞x he probability that the random variable takes a value less than or equal to x. F(x) is non‑decreasing, with F(x) → 0 as x → −∞ and F(x) → 1 as x → ∞.
累积分布函数 F(x) = P(X ≤ x) = ∫−∞x f(t) dt。它给出了随机变量取值小于或等于 x 的概率。F(x) 是非递减的，且当 x → −∞ 时 F(x) → 0，当 x → ∞ 时 F(x) → 1。
b) = F(b) − F(a). Also, the PDF can be recovered from the CDF by differentiation: f(x) = F'(x) at points where F is differentiable.
根据概率密度函数，我们可以求出诸如 P(a < X ≤ b) = F(b) − F(a) 的概率。同时，概率密度函数可以通过对累积分布函数求导得到：在 F 可微的点处，f(x) = F'(x)。
4. Finding the Median and Quartiles | 求中位数与四分位数
The median m of a continuous distribution satisfies P(X ≤ m) = 0.5, i.e., ∫−∞m f(x) dx = 0.5. Similarly, the lower quartile Q₁ satisfies ∫−∞Q₁ f(x) dx = 0.25, and the upper quartile Q₃ satisfies ∫−∞Q₃ f(x) dx = 0.75.
连续分布的中位数 m 满足 P(X ≤ m) = 0.5，即 ∫−∞m f(x) dx = 0.5。类似地，下四分位数 Q₁ 满足 ∫−∞Q₁ f(x) dx = 0.25，上四分位数 Q₃ 满足 ∫−∞Q₃ f(x) dx = 0.75。
ven probability. If the distribution is defined on a finite interval [a, b], the lower limit of integration becomes a.
这些值通过解累积分布方程 F(x) = 给定的概率来求得。若分布定义在有限区间 [a, b] 上，积分的下限即为 a。
5. Mode of a Continuous Distribution | 连续分布的众数
at maximises f(x) within its domain. It may be found by differentiating f(x) and setting f'(x) = 0, checking that the second derivative is negative or using interval boundaries.
连续随机变量的众数是定义域内使 f(x) 取得最大值的 x 值。它可以通过对 f(x) 求导并令 f'(x) = 0 来寻找，并检验二阶导数为负或利用区间端点。
A distribution may be unimodal, bimodal, or multimodal. If f(x) is strictly increasing on the interval, the mode is at the right endpoint; if strictly decreasing, the mode is at the left endpoint.
分布可能是单峰的、双峰的或多峰的。如果 f(x) 在区间上严格递增，众数位于右端点；若严格递减，则众数位于左端点。
6. Expectation E(X) and E(g(X)) | 期望 E(X) 与 E(g(X))
ous random variable X is E(X) = ∫ x f(x) dx over the entire domain. It represents the long‑run average of X.
连续随机变量 X 的期望值（均值）为 E(X) = ∫ x f(x) dx（在整个定义域上）。它代表了 X 的长期平均值。
∫ g(x) f(x) dx. Common uses include E(X²) for finding variance: Var(X) = E(X²) − [E(X)]².
对于函数 g(X)，期望为 E(g(X)) = ∫ g(x) f(x) dx。常见的用法包括求取方差所需的 E(X²)：Var(X) = E(X²) − [E(X)]²。
7. Variance and Standard Deviation | 方差与标准差
where μ = E(X). Equivalently, Var(X) = E(X²) − μ². The standard deviation is σ = √Var(X).
方差 Var(X) = ∫ (x − μ)² f(x) dx，其中 μ = E(X)。等价地，Var(X) = E(X²) − μ²。标准差为 σ = √Var(X)。
Variance measures the spread of the distribution. A larger variance indicates that values are more dispersed around the mean.
方差衡量分布的分散程度。方差越大，表明数值在均值周围越分散。
8. Finding Probabilities Using the PDF | 利用概率密度函数求概率
To compute P(a < X < b), integrate f(x) from a to b. If the distribution is piecewise, ensure the limits lie within the non‑zero part of the PDF.
要计算 P(a < X < b)，对 f(x) 从 a 到 b 积分。如果分布是分段函数，确保积分限落在概率密度函数的非零部分内。
/4 [(0.5 − 0.125/3) − (−0.5 + 0.125/3)] = 3/4 [2(0.5 − 0.04167)] = 3/4 × 0.91666 = 0.6875.
例：对于 f(x) = 3/4 (1 − x²) 在 [−1, 1] 上，求 P(−0.5 < X < 0.5)。从 −0.5 到 0.5 积分：∫ 3/4 (1 − x²) dx = 3/4 [x − x³/3] 从 −0.5 到 3/4 [(0.5 − 0.125/3) − (−0.5 + 0.125/3)] = 3/4 [2(0.5 − 0.04167)] = 3/4 × 0.91666 = 0.6875。
9. Linear Transformations of a Continuous Random Variable | 连续随机变量的线性变换
If Y = aX + b, where X has PDF fX nd Var(Y) = a² Var(X). The PDF of Y can be found using the change of variable technique: fY(y) = fX((y − b)/a) × |1/a|.
若 Y = aX + b，其中 X 的概率密度函数为 fX(x)，则 E(Y) = aE(X) + b，且 Var(Y) = a² Var(X)。Y 的概率密度函数可通过变量替换法求得：fY(y) = fX((y − b)/a) × |1/a|。
tribution is derived from a general normal distribution via standardisation Z = (X − μ)/σ.
这在通过标准化 Z = (X − μ)/σ 从一般正态分布导出标准正态分布时尤其有用。
10. The Uniform Distribution as a Simple PDF | 均匀分布作为一种简单的概率密度函数
b. This is the simplest continuous PDF and serves as a foundation for understanding rectangular areas.
区间 [a, b] 上的连续均匀分布其概率密度函数为 f(x) = 1/(b − a)，a ≤ x ≤ b。这是最简单的连续型概率密度函数，为理解矩形面积奠定了基础。
(a + b)/2, variance is (b − a)²/12, and the CDF is F(x) = (x − a)/(b − a) for x in [a, b].
其均值为 (a + b)/2，方差为 (b − a)²/12，累积分布函数为 F(x) = (x − a)/(b − a)（x 在 [a, b] 内）。
11. Modeling Real‑World Scenarios with PDFs | 用概率密度函数建模实际场景
ors, and natural phenomena. For example, the exponential distribution with PDF f(x) = λ e⁻λx (x ≥ 0) models the time between events in a Poisson process.
概率密度函数被广泛用于建模等待时间、测量误差和自然现象。例如，指数分布的概率密度函数为 f(x) = λ e⁻λx（x ≥ 0），用于建模泊松过程中事件发生的时间间隔。
In Edexcel exam questions, you may be asked to identify a suitable distribution or calculate parameters such as mean and standard deviation from a given PDF, and then interpret the results in context.
在 Edexcel 考题中，可能要求识别合适的分布，或根据给定的概率密度函数计算均值和标准差等参数，并结合情境解释结果。
12. Common Pitfalls and Exam Tips | 常见误区与备考提示
he PDF integrates to 1 when finding unknown constants, or they confuse P(X = a) with a non‑ emember, for continuous distributions, probability is always an area, never a point value.
学生在求未知常数时常常忘记检验概率密度函数的积分为1，或者误以为 P(X = a) 为非零值。请记住，对于连续分布，概率始终是一个面积，而非某点处的值。
When calculating the median, ensure you use the correct limits. If the PDF is piecewise, determine which section contains the median before integrating. Also, double‑check arithmetic in variance calculations, as errors in E(X²) are common.
计算中位数时，确保使用正确的上下限。如果概率密度函数是分段函数，在进行积分前先确定中位数位于哪个区间。此外，方差计算中的 E(X²) 容易出错，务必仔细核验算术。
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