📚 Investigation 1 - Probability density functions | 探究一：概率密度函数
In this investigation, we explore probability density functions (PDFs), a fundamental concept for modelling continuous random variables. From heights to measurement errors, PDFs help us quantify uncertainty in continuous settings and form the backbone of statistical inference in the IB curriculum.
在本探究中，我们将探索概率密度函数（PDF），这是模拟连续随机变量的基本概念。从身高到测量误差，PDF帮助我们量化连续情况下的不确定性，并构成IB课程中统计推断的基石。
1. Introduction to Continuous Random Variables | 连续随机变量简介
A continuous random variable can take any value within an interval or union of intervals. Unlike discrete variables, which have a probability mass function (PMF) that assigns probabilities to individual outcomes, continuous variables require a probability density function to describe their distribution because the probability of any single exact value is zero.
连续随机变量可以取某个区间或区间并集内的任何值。与离散变量不同，离散变量通过概率质量函数（PMF）为单个结果分配概率，而连续变量需要概率密度函数来描述其分布，因为任何一个精确值的概率为零。
For example, the exact time a runner finishes a marathon is a continuous variable (it could be 2:08:32.185...), whereas the number of goals in a football match is discrete. The height of an adult chosen at random also behaves as a continuous variable.
例如，跑者完成马拉松的确切时间是一个连续变量（可能是2:08:32.185...），而足球比赛中的进球数则是离散的。随机选出的成年人身高也是一个连续变量。
2. What is a Probability Density Function? | 什么是概率密度函数？
that X lies in the interval [a, b] is the area under the curve: P(a ≤ X ≤ b) = ∫ab f(x) dx.
概率密度函数 f(x) 是一个非负函数，描述连续随机变量 X 取值在 x 附近的可能性相对大小。X 落在区间 [a, b] 内的概率是曲线下的面积：P(a ≤ X ≤ b) = ∫ₐᵇ f(x) dx。
f(x) itself is not a probability; it is a density. For a single point c, P(X = c) = ∫cc f(x) dx = 0, which is always true for a continuous random variable.
理解 f(x) 本身不是概率，而是密度，这一点至关重要。对于单点 c，P(X = c) = ∫ₓᶜ f(x) dx = 0，这对任何连续随机变量都成立。
3. Properties of a PDF | 概率密度函数的性质
Every valid PDF must satisfy two essential conditions: (i) f(x) ≥ 0 for all real x, ensuring no negative likelihoods; (ii) the total area under the curve equals 1, i.e., ∫−∞∞ 1. These properties guarantee that the function represents a legitimate probability model.
每个有效的PDF必须满足两个基本条件：(i) 对所有实数 x，f(x) ≥ 0，确保没有负的可能性；(ii) 曲线下的总面积为1，即 ∫₋∞⁺∞ f(x) dx = 1。这些性质保证了函数表示一个合法的概率模型。
If a proposed function violates non-negativity or its integral over the entire real line is not 1, it cannot serve as a PDF. We sometimes need to find a normalisation constant to ensure the second condition holds.
如果提议的函数违反非负性或其在整条实数线上的积分不等于1，它就不能作为PDF。有时我们需要找到一个归一化常数以确保第二个条件成立。
4. The Cumulative Distribution Function (CDF) | 累积分布函数
obability that X is less than or equal to a particular value x: F(x) = P(X ≤ x) = ∫−∞x f(t) dt. It is a non-decreasing function that starts at 0 and approaches 1 as x → ∞.
累积分布函数 F(x) 返回 X 小于或等于某个特定值 x 的概率：F(x) = P(X ≤ x) = ∫₋∞ˣ f(t) dt。它是一个非递减函数，从0开始，当 x → ∞ 时趋近于1。
For a continuous random variable, f(x) is obtained by differentiating the CDF: f(x) = F' s relationship makes it easy to move between the density and the cumulative probabilities.
对于连续随机变量，f(x) 可以通过对CDF求导得到：f(x) = F'(x)，前提是导数存在。这种关系使得在密度函数与累积概率之间转换变得很容易。
5. Calculating Probabilities Using PDFs | 使用概率密度函数计算概率
and 0 otherwise. To find P(0.3 ≤ T ≤ 0.7), we integrate f(t) over [0.3, 0.7]: ∫0.30.7 2t dt = [t²]0.30.7 = 0.49 − 0.09 = 0.40.
假设连续随机变量 T 的 PDF 为 f(t) = 2t，0 ≤ t ≤ 1，否则为0。为了求 P(0.3 ≤ T ≤ 0.7)，我们在 [0.3, 0.7] 上积分：∫₀.₃⁰.⁷ 2t dt = [t²]₀.₃⁰.⁷ = 0.49 − 0.09 = 0.40。
This example illustrates that computing a probability is equivalent to finding the area of a trapezoid under a straight line
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