Probability Distributions | 概率分布

📚 Probability Distributions | 概率分布

In A-Level Edexcel Mathematics, probability distributions formalise how probabilities are assigned to outcomes of a random variable. A random variable maps outcomes of a random experiment to numerical values, and its distribution gives the likelihood of each possible value. Mastering notation, expected value and variance is essential for modelling binomial and Poisson scenarios.

在 A-Level Edexcel 数学中，概率分布为描述随机性提供了规范语言。随机变量将随机试验的结果映射到数值，其分布给出每个可能取值的概率。掌握记号、期望值与方差，对建立二项分布和泊松分布模型至关重要。

1. Random Variables and Sample Spaces | 随机变量与样本空间

A random variable, usually written X, is a rule that assigns a real number to each outcome of a random experiment. For example, if a coin is tossed twice, X could be the number of heads, taking values 0, 1 or 2.

随机变量通常写作 X，它是将随机试验的每个结果对应到一个实数的规则。例如，若抛硬币两次，X 可以表示正面朝上的次数，取值为 0、1 或 2。

If the set of possible values is finite or countable, X is discrete. If X can take any value in an interval, it is continuous. Edexcel questions often expect you to identify the type of variable before choosing a method.

如果可能取值的集合是有限或可数的，则 X 是离散型；如果 X 可以取区间中的任意值，则它是连续型。Edexcel 题目通常要求先判断变量类型，再选择方法。

2. Discrete Probability Distributions and the Probability Mass Function | 离散概率分布与概率质量函数

For a discrete random variable X, the probability mass function (PMF) lists all possible values x and the probabilities P(X = x). The two necessary conditions are: 0 ≤ P(X = x) ≤ 1 for every x, and the sum of all probabilities equals 1.

对于离散随机变量 X，概率质量函数 (PMF) 列出所有可能取值 x 以及概率 P(X = x)。必须满足两个条件：每个 x 都有 0 ≤ P(X = x) ≤ 1，且所有结果的概率之和等于 1。

A distribution can be displayed as a table. For example, if X is the number of heads when a fair coin is tossed twice, then X can be 0, 1 or 2 with probabilities 1/4, 1/2 and 1/4.

分布可以用表格表示。例如，若 X 表示抛两次公平硬币正面朝上的次数，则 X 可取 0、1、2，概率分别为 1/4、1/2、1/4。

3. Cumulative Distribution Function | 累积分布函数

The cumulative distribution function (CDF) is F(x) = P(X ≤ x). For a discrete variable, F(x) is obtained by summing the probabilities of all values less than or equal to x.

累积分布函数 (CDF) 为 F(x) = P(X ≤ x)。对于离散变量，F(x) 通过将所有小于等于 x 的取值概率相加得到。

Key properties are: F(x) is non-decreasing, 0 ≤ F(x) ≤ 1, F(x) tends to 0 as x → −∞ and tends to 1 as x → ∞. The probability of an interval can be found as P(a < X ≤ b) = F(b) − F(a).

关键性质包括：F(x) 单调不减，0 ≤ F(x) ≤ 1，当 x → −∞ 时 F(x) 趋于 0，当 x → ∞ 时趋于 1。区间概率可表示为 P(a < X ≤ b) = F(b) − F(a)。

Edexcel papers often ask you to construct or interpret a cumulative distribution table. Make sure cumulative probabilities are found by successive addition, not by multiplying individual probabilities.

Edexcel 试卷常要求构建或解读累积分布表。务必通过逐项累加得到累积概率，而不是将单个概率相乘。

4. Expected Value E(X) | 期望值 E(X)

The expected value, or mean, of a discrete random variable is E(X) = Σ x P(X = x). It measures the long-run average outcome if the experiment were repeated many times.

离散随机变量的期望值（或均值）为 E(X) = Σ x P(X = x)。它衡量在大量重复试验中的长期平均结果。

E(X) = Σ x P(X = x)

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