📚 Probability Distributions | 概率分布
In IB Mathematics, probability distributions provide a framework for modelling how random variables behave. This article explores discrete and continuous distributions, covering essential concepts from binomial trials to normal curves, enabling you to solve core problems with confidence.
在IB数学中，概率分布为随机变量的行为建模提供了框架。本文探讨离散和连续分布，涵盖从伯努利试验到正态曲线的基本概念，帮助你自信地解决核心问题。
1. Random Variables and Types | 随机变量及其类型
A random variable (RV) is a variable whose numerical value is determined by the outcome of a random experiment. We distinguish two principal types: discrete RVs, which take a countable number of distinct values, and continuous RVs, which can assume any value within an interval.
随机变量（RV）是由随机试验结果决定其数值的变量。主要分为两类：离散随机变量（取可数的、互不相同的值）和连续随机变量（可在某个区间内取任意值）。
Discrete examples include the number of heads when flipping three coins; continuous examples include the exact height of a randomly chosen student. The mathematical treatment of these two types differs fundamentally – probabilities are summed for discrete RVs and integrated for continuous ones.
离散的例子如抛三枚硬币时正面朝上的次数；连续的例子如随机选取一名学生的准确身高。这两类变量的数学处理方式根本不同——离散随机变量通过求和计算概率，连续随机变量通过积分计算概率。
2. Discrete Probability Distributions | 离散概率分布
A discrete probability distribution lists all possible values x of a discrete RV X together with their probabilities P(X = x). Each probability lies between 0 and 1, and the sum over all possible x equals 1.
离散概率分布列出离散随机变量 X 的所有可能取值 x 及其概率 P(X = x)。每个概率在 0 到 1 之间，且所有可能取值的概率之和等于 1。
For example, a fair six-sided die yields the uniform distribution: P(X = k) = 1/6 for k = 1, 2, … y a table or a function and is the foundation for computing expected values and variances.
例如，一枚均匀的六面骰子给出均匀分布：P(X = k) = 1/6，k = 1, 2, …, 6。该分布可用表格或函数表示，是计算期望值和方差的基础。
3. Expected Value (Mean) of a Discrete Random Variable | 离散随机变量的期望值
The expected value E(X), denoted by the Greek letter μ (mu), is the long‑run average of a random variable. For a discrete RV it is calculated as E(X) = Σ x · P(X = x), where the sum runs over all possible values of X.
期望值 E(X)，用希腊字母 μ（均）表示，是随机变量的长期平均值。对于离散随机变量，计算公式为 E(X) = Σ x · P(X = x)，求和遍历 X 的所有可能取值。
The die example has E(X) = 1·(1/6) + 2·(1/6) + … + 6·(1/6) = 3.5. Note that the expected value need not be an actual attainable value – you can never roll a 3.5, but it represents the average over many trials.
骰子例子的 E(X) = 1·(1/6) + 2·(1/6) + … + 6·(1/6) = 3.5。注意期望值不一定是实际可能取到的值——你永远掷不出 3.5，但它代表多次试验的平均结果。
4. Variance and Standard Deviation | 方差与标准差
Variance Var(X) measures the dispersion of a distribution – how far the values are spread from the mean. It is defined as the expected squared deviation: Var(X) = E[(X – μ)2] = Σ (x – μ)2 · P(X = x).
方差 Var(X) 衡量分布的离散程度——取值距离均值的散布程度。其定义为偏差平方的期望：Var(X) = E[(X – μ)2] = Σ (x – μ)2 · P(X = x)。
A computationally convenient formula is Var(X) = E(X2) – [E(X)]2, where E(X2) = Σ x2 · P(X = x). The standard deviation σ is the square root of the variance, σ = √Var(X), and shares the same units as X.
一个便于计算的公式为 Var(X) = E(X2) – [E(X)]2，其中 E(X2) = Σ x2 · P(X = x)。标准差 σ 是方差的平方根，σ = √Var(X)，其单位与 X 相同。
5. The Binomial Distribution | 二项分布
A binomial distribution arises when we count the number of successes in a fixed number n of independent trials, each having two outcomes (success/failure) with constant success probability p. We write X ~ B(n, p).
当我们在固定次数 n 的独立试验中计数成功次数，且每次试验仅有成功或失败两种结果，成功概率 p 保持不变时，就得到二项分布。记为 X ~ B(n, p)。
The probability mass function is P(X = k) = C(n, k) pk (1–p)n–k, where C(n, k) = n!/(k!(n–k)!) is the binomial coefficient, giving the number of ways to choose k successes out of n trials.
其概率质量函数为 P(X = k) = C(n, k) pk (1–p)n–k，其中二项式系数 C(n, k) = n!/(k!(n–k)!) 表示从 n 次试验中选出 k 次成功的方法数。
The four key conditions for a binomial setting are: fixed number of trials n, each trial produces exactly two possible results, the probability of success p is the same on every trial, and trials are independent.
二项分布成立的四个关键条件为：试验次数 n 固定，每次试验只有两种可能结果，每次试验的成功概率 p 相同，且各次试验相互独立。
6. Mean and Variance of a Binomial Distribution | 二项分布的均值与方差
For X ~ B(n, p), the expected value is simply μ = n p, and the variance is Var(X) = n p(1–p). The standard deviation follows as σ = √[n p(1–p)].
对于 X ~ B(n, p)，期望值为 μ = n p，方差为 Var(X) = n p(1–p)。标准差为 σ = √[n p(1–p)]。
These compact formulas avoid the need to sum over all k and clearly show that the spread is largest when p = 0.5, for a given n.
这些简明的公式避免了对所有 k 求和，并清楚地表明，在给定的 n 下，当 p = 0.5 时分布最分散。
7. Continuous Random Variables | 连续随机变量
A continuous random variable can take any value in a real interval. Because there are infinitely many possible values, the probability of any single exact value is zero; instead we consider the probability that the variable falls within a certain interval.
连续随机变量可以在实数区间内取任意值。由于存在无穷多个可能取值，取任一特定值的概率为零；因此我们常考虑变量落在某个区间内的概率。
The distribution is described by a probability density function (pdf), where the area under the curve between two points gives the probability: P(a ≤ X ≤ b) = ∫ab f(x) dx.
分布由概率密度函数（pdf）描述，曲线下两点之间的面积即为概率：P(a ≤ X ≤ b) = ∫ab f(x) dx。
8. Probability Density Function (pdf) | 概率密度函数
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