📚 IB Mathematics: Fundamental Concepts of Probability Distributions | IB数学：概率分布基础概念解析
Probability distributions are the mathematical backbone of statistics and are essential for the IB Mathematics curriculum. They describe how the probabilities of different outcomes are spread across the possible values of a random variable.
概率分布是统计学的数学基石，也是IB数学课程的核心内容。它描述了随机变量所有可能取值的概率是如何分布的。
1. Random Variables | 随机变量
A random variable is a variable whose possible values are numerical outcomes of a random phenomenon. It is usually denoted by an uppercase letter, such as X, while its observed values are written in lowercase, such as x.
随机变量是将随机现象的数值结果赋值为变量的数学工具。通常用大写字母如X表示随机变量，其具体观测值用小写字母如x表示。
For example, when rolling a fair six-sided die, let X be the number shown. Then X can take values 1, 2, 3, 4, 5, or 6, each with probability 1/6.
例如，掷一枚均匀六面骰子时，令X表示朝上的点数。那么X可以取1、2、3、4、5或6，每个取值的概率都是1/6。
Random variables are classified into two main types: discrete and continuous.
随机变量主要分为两类：离散型随机变量和连续型随机变量。
2. Discrete vs Continuous Distributions | 离散与连续分布
A discrete random variable takes a countable number of distinct values. Examples include the number of heads in three coin tosses, or the score on a single exam question.
离散型随机变量只取可数的、彼此分离的值。例如三次抛硬币中正面出现的次数，或一道题目的得分。
A continuous random variable takes any value within an interval. Examples include the height of a student, the time taken to complete a task, or the temperature at noon.
连续型随机变量可以取某个区间内的任意值。例如学生的身高、完成一项任务所需的时间，或正午的气温。
For a discrete distribution, probabilities are assigned to each individual value. For a continuous distribution, probabilities are assigned to intervals because the probability of any single exact value is zero.
对于离散分布，概率分配给每一个具体取值；而连续分布中，概率分配给区间，因为单一精确值的概率为零。
3. Probability Mass Function (PMF) | 概率质量函数
The probability mass function (PMF) of a discrete random variable X gives the probability that X equals a specific value x. It is written as P(X = x) = p(x).
离散型随机变量X的概率质量函数（PMF）给出X等于某一具体值x的概率，记作P(X = x) = p(x)。
A PMF must satisfy two conditions:
一个概率质量函数必须满足两个条件：
- P(X = x) ≥ 0 for every possible value x.
- The sum over all possible values equals 1: Σ P(X = x) = 1.
- 对每个可能的x，有P(X = x) ≥ 0。
- 所有可能取值的概率之和等于1：Σ P(X = x) = 1。
For a fair die, the PMF is P(X = x) = 1/6 for x = 1, 2, 3, 4, 5, 6.
对于均匀骰子，其概率质量函数为：当x = 1, 2, 3, 4, 5, 6时，P(X = x) = 1/6。
4. Probability Density Function (PDF) | 概率密度函数
For a continuous random variable, the probability density function (PDF), denoted f(x), describes the relative likelihood of the random variable taking values near x.
对于连续型随机变量，概率密度函数（PDF）记作f(x)，它描述了随机变量在x附近取值的相对可能性。
Key properties of a PDF are:
概率密度函数的主要性质包括：
- f(x) ≥ 0 for all x.
- The total area under the curve is 1: ∫ f(x) dx = 1 over the entire range.
- The probability that X lies in an interval [a, b] is the area under the curve from a to b: P(a ≤ X ≤ b) = ∫ab f(x) dx.
- 对于所有x，f(x) ≥ 0。
- 曲线下的总面积为1：在整个取值范围内∫ f(x) dx = 1。
- X落在区间[a, b]内的概率等于曲线从a到b的面积：P(a ≤ X ≤ b) = ∫ab f(x) dx。
Notice that for a continuous variable, P(X = a) = 0, so including or excluding endpoints does not change the probability.
注意，对于连续型变量，P(X = a) = 0，因此区间端点是否包含在内不影响概率值。
5. Cumulative Distribution Function (CDF) | 累积分布函数
The cumulative distribution function (CDF) of a random variable X, denoted F(x), gives the probability that X is less than or equal to x:
随机变量X的累积分布函数（CDF）记作F(x)，表示X小于或等于x的概率：
F(x) = P(X ≤ x)
For a discrete variable, the CDF is obtained by summing the PMF values up to x:
对于离散型变量，累积分布函数通过对所有不超过x的概率质量函数值求和得到：
F(x) = Σt ≤ x P(X = t)
For a continuous variable, the CDF is the integral of the PDF from −∞ to x:
对于连续型变量，累积分布函数是概率密度函数从−∞到x的积分：
F(x) = ∫−∞x f(t) dt
The CDF is always non-decreasing and approaches 0 as x → −∞ and 1 as x → +∞.
累积分布函数总是单调不减的，当x → −∞时趋于0，当x → +∞时趋于1。
6. Expected Value | 期望值
The expected value (or mean) of a random variable X measures its central tendency. It is denoted E(X) or μ.
随机变量X的期望值（或均值）衡量其中心趋势，记作E(X)或μ。
For a discrete random variable with PMF P(X = x), the expected value is:
对于具有概率质量函数P(X = x)的离散型随机变量，期望值为：
E(X) = Σ x · P(X = x)
For a continuous random variable with PDF f(x), the expected value is:
对于具有概率密度函数f(x)的连续型随机变量，期望值为：
E(X) = ∫ x · f(x) dx
The expected value is not necessarily a value that X can actually take; it is a weighted average over all outcomes.
期望值不一定是X实际能取的某个值，而是对所有结果按概率加权的平均值。
7. Variance and Standard Deviation | 方差与标准差
Variance measures the spread of a distribution around the mean. It is defined as the expected value of the squared deviation from the mean:
方差衡量分布围绕均值的离散程度，定义为离均差平方的期望值：
Var(X) = E[(X − μ)²] = E(X²) − [E(X)]²
The standard deviation is the square root of the variance:
标准差是方差的正平方根：
σ = √Var(X)
A larger variance or standard deviation indicates that the values are more spread out from the mean.
方差或标准差越大，说明取值距离均值越分散。
In IB problems, you are often asked to compute E(X²) first, then use the shortcut formula above to find Var(X).
在IB题目中，经常需要先计算E(X²)，再使用上述简化公式求Var(X)。
8. Binomial Distribution | 二项分布
The binomial distribution models the number of successes in n independent trials, where each trial has the same probability of success p. It is denoted X ~ B(n, p).
二项分布用于建模n次独立试验中成功的次数，每次试验的成功概率均为p，记作X ~ B(n, p)。
The probability of exactly k successes is given by:
恰好取得k次成功的概率为：
P(X = k) = ⁿCₖ · pᵏ · (1 − p)ⁿ⁻ᵏ
Here ⁿCₖ is the binomial coefficient, also written as C(n, k) or “n choose k”.
其中ⁿCₖ是二项系数，也可写作C(n, k)或“n选k”。
The mean and variance of a binomial distribution are:
二项分布的均值与方差分别为：
E(X) = np, Var(X) = np(1 − p)
This distribution is widely used in quality control, biology, and opinion polls.
二项分布在质量控制、生物学和民意调查等领域有广泛的应用。
9. Normal Distribution | 正态分布
The normal distribution is the most important continuous distribution. It has a symmetric bell-shaped curve and is characterised by its mean μ and standard deviation σ. We write X ~ N(μ, σ²).
正态分布是最重要的连续型分布。它的曲线呈对称钟形，由均值μ和标准差σ决定，记作X ~ N(μ, σ²)。
The total area under the normal curve is 1, and about 68% of values lie within one standard deviation of the mean, 95% within two, and 99.7% within three.
正态曲线下的总面积为1。约68%的取值落在均值的一个标准差范围内，约95%落在两个标准差范围内，约99.7%落在三个标准差范围内。
The normal distribution is denoted by:
正态分布的密度函数为：
f(x) = (1 / (σ√(2π))) · e^(−(x−μ)² / (2σ²))
In IB examinations, you will usually use a GDC or a normal probability table to find probabilities, rather than integrating the density function directly.
在IB考试中，通常使用图形计算器（GDC）或正态分布表来求概率，而不是直接对密度函数积分。
10. Standardising and the z-Score | 标准化与z分数
Any normal distribution can be transformed into the standard normal distribution N(0, 1) using the z-score:
任何正态分布都可以通过z分数变换为标准正态分布N(0, 1)：
z = (X − μ) / σ
The z-score tells us how many standard deviations a value X is above or below the mean.
z分数表示某个取值X高于或低于均值多少个标准差。
If X ~ N(μ, σ²), then Z ~ N(0, 1). This allows us to use a single standard table for all normal probability calculations.
如果X ~ N(μ, σ²)，则Z ~ N(0, 1)。这样我们就可以用同一张标准正态分布表处理所有正态分布概率计算。
For example, if X has mean 50 and standard deviation 5, the value X = 60 corresponds to z = (60 − 50)/5 = 2, which is two standard deviations above the mean.
例如，若X的均值为50，标准差为5，则X = 60对应的z = (60 − 50)/5 = 2，说明它高于均值两个标准差。
11. Discrete vs Continuous: Working with Probabilities | 离散与连续：概率计算的区别
For a discrete distribution, probabilities are found by adding individual PMF values. For example, P(X ≤ 3) = P(X = 1) + P(X = 2) + P(X = 3).
对于离散分布，概率通过对个别概率质量函数值求和得到。例如，P(X ≤ 3) = P(X = 1) + P(X = 2) + P(X = 3)。
For a continuous distribution, probabilities are found by integrating the PDF over the interval. For example, P(a ≤ X ≤ b) = ∫ab f(x) dx.
对于连续分布，概率通过对概率密度函数在区间上积分得到。例如，P(a ≤ X ≤ b) = ∫ab f(x) dx。
Because continuous probabilities are areas under a curve, P(X ≤ c) = P(X < c), and P(X > a) = 1 − P(X ≤ a).
由于连续概率是曲线下的面积，所以P(X ≤ c) = P(X < c)，且P(X > a) = 1 − P(X ≤ a)。
Always check whether the random variable is discrete or continuous before choosing the correct method for probability calculation.
在计算概率之前，一定要先判断随机变量是离散型还是连续型，从而选择正确的方法。
12. Common Exam Pitfalls | 常见考试易错点
Students often confuse the PMF and the CDF. Remember: the PMF gives P(X = x), while the CDF gives P(X ≤ x).
学生常常混淆概率质量函数和累积分布函数。请记住：PMF给出P(X = x)，CDF给出P(X ≤ x)。
Another common mistake is using the binomial formula when trials are not independent, or when n and p are not clearly identified.
另一个常见错误是在各次试验不独立时误用二项分布公式，或者没有明确识别出n和p。
For normal distribution questions, always standardise using z = (X − μ)/σ before using a table, and check whether the question gives a one-tailed or two-tailed probability.
对于正态分布题目，使用表格前务必先通过z = (X − μ)/σ标准化，并判断题目求的是单尾还是双尾概率。
Finally, do not forget that the total probability is always 1. If your computed probabilities do not sum to 1, go back and check your work.
最后，不要忘记总概率始终为1。如果你算出的概率之和不等于1，请回头检查你的计算过程。
Published by TutorHao | IB Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导