📚 IB Mathematics: Continuous Random Variables and Their Distributions | IB数学：连续随机变量及其分布要点
In probability and statistics, a continuous random variable is a random variable with a continuous range of values. Unlike discrete random variables, which take isolated values, continuous random variables can take any value within an interval. In IB Mathematics, understanding their distributions is essential for solving realistic problems.
在概率与统计中，连续随机变量是指取值包含连续区间的随机变量。与取孤立值的离散随机变量不同，连续随机变量可以在某个区间内取任意值。在IB数学中，掌握其分布是解决实际问题的基础。
1. Definition and Conditions | 定义与条件
A continuous random variable X is defined by a probability density function (pdf) f(x) such that P(a ≤ X ≤ b) equals the area under the curve f(x) from a to b. The two fundamental conditions are that f(x) ≥ 0 for all x, and the total area under the curve is 1.
连续随机变量 X 由概率密度函数 f(x) 定义，使得 P(a ≤ X ≤ b) 等于曲线 f(x) 在 a 到 b 之间的面积。两个基本条件是：对所有 x，f(x) ≥ 0；并且曲线下总面积为 1。
∫-∞∞ f(x) dx = 1
For an interval [a, b], the probability is given by the integral of the pdf over that interval. Note that P(X = c) = 0 for any single point c.
对于区间 [a, b]，概率等于概率密度函数在该区间上的积分。注意任意单点 c 的概率 P(X = c) = 0。
P(a ≤ X ≤ b) = ∫ab f(x) dx
2. Cumulative Distribution Function | 累积分布函数
The cumulative distribution function F(x) gives the probability that X is less than or equal to x.
累积分布函数 F(x) 表示 X 小于或等于 x 的概率。
F(x) = P(X ≤ x) = ∫-∞x f(t) dt
Key properties:
关键性质：
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F'(x) = f(x) where f is continuous.
在 f 连续处，F'(x) = f(x)。
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F(-∞) = 0 and F(∞) = 1.
F(-∞) = 0 且 F(∞) = 1。
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P(a < X < b) = F(b) - F(a).
P(a < X < b) = F(b) - F(a)。
3. Expectation and Variance | 期望与方差
The expected value (mean) of X is defined by the integral of x times the pdf.
X 的期望值（均值）定义为 x 乘以概率密度函数的积分。
E[X] = ∫-∞∞ x f(x) dx
The variance is the expected squared deviation from the mean.
方差是偏离均值的平方的期望。
Var(X) = E[(X – μ)²] = ∫-∞∞ (x – μ)² f(x) dx
An equivalent formula is Var(X) = E[X²] – μ². For linear transformations, E[aX + b] = aE[X] + b and Var(aX + b) = a²Var(X).
等效公式为 Var(X) = E[X²] – μ²。对线性变换，有 E[aX + b] = aE[X] + b 和 Var(aX + b) = a²Var(X)。
4. Median, Quartiles and Mode | 中位数、四分位数与众数
The median m solves F(m) = 0.5. The lower quartile q₁ solves F(q₁) = 0.25, and the upper quartile q₃ solves F(q₃) = 0.75.
中位数 m 满足 F(m) = 0.5；下四分位数 q₁ 满足 F(q₁) = 0.25；上四分位数 q₃ 满足 F(q₃) = 0.75。
The mode is the value of x where f(x) is maximal. These measures of location are widely used in IB statistics and data analysis.
众数是使 f(x) 达到最大值的 x 值。这些位置度量在IB统计和数据分析中应用广泛。
5. Uniform Distribution | 均匀分布
A continuous uniform distribution U(a, b) gives equal probability density to all values in the interval [a, b].
连续均匀分布 U(a, b) 赋予区间 [a, b] 内所有取值相同的概率密度。
f(x) = 1/(b – a) for a ≤ x ≤ b
The cumulative distribution function is:
其累积分布函数为：
F(x) = (x – a)/(b – a) for a ≤ x ≤ b
Its mean and variance are:
它的均值与方差为：
E[X] = (a + b)/2， Var(X) = (b – a)²/12
Uniform distributions are used to model equally likely outcomes over a continuous range.
均匀分布用于模拟连续范围内等可能的结果。
6. Exponential Distribution | 指数分布
The exponential distribution has pdf f(x) = λe^(−λx) for x ≥ 0, where λ > 0 is the rate parameter.
指数分布的概率密度函数为 f(x) = λe^(−λx)（x ≥ 0），其中 λ > 0 是速率参数。
f(x) = λe^(−λx) for x ≥ 0
The mean and variance are E[X] = 1/λ and Var(X) = 1/λ².
其均值与方差为 E[X] = 1/λ，Var(X) = 1/λ²。
The CDF is F(x) = 1 − e^(−λx). A key property is lack of memory:
其累积分布函数为 F(x) = 1 − e^(−λx)。一个关键性质是无记忆性：
P(X > s + t | X > s) = P(X > t)
7. Normal Distribution | 正态分布
The normal distribution N(μ, σ²) is the most important continuous distribution. Its pdf is bell-shaped and symmetric about the mean μ.
正态分布 N(μ, σ²) 是最重要的连续分布。它的概率密度函数呈钟形，且关于均值 μ 对称。
f(x) = (1/(σ√(2π))) e^(−(x−μ)²/(2σ²))
Its mean is μ and its variance is σ². The empirical rule states that about 68%, 95% and 99.7% of data lie within 1, 2 and 3 standard deviations from the mean.
其均值为 μ，方差为 σ²。经验法则表明：约 68%、95%、99.7% 的数据分别落在距均值 1、2、3 个标准差范围内。
8. Standard Normal Distribution | 标准正态分布
Any normal variable X can be standardized to Z = (X − μ)/σ, which follows N(0, 1).
任何正态变量 X 都可以标准化为 Z = (X − μ)/σ，它服从 N(0, 1)。
Z = (X − μ)/σ
Probabilities for Z are found using the standard normal table or a calculator. For example, P(Z < z) = Φ(z). By symmetry, P(Z ≤ −z) = 1 − Φ(z).
Z 的概率可通过标准正态分布表或计算器查找。例如，P(Z < z) = Φ(z)。由对称性，P(Z ≤ −z) = 1 − Φ(z)。
To find X from a given probability, use X = μ + zσ, where z is the corresponding standard normal quantile.
若已知概率反求 X，使用 X = μ + zσ，其中 z 是对应的标准正态分位数。
9. Normal Approximation to the Binomial | 正态近似二项分布
When n is large, the binomial distribution Bin(n, p) can be approximated by N(np, np(1 − p)).
当 n 很大时，二项分布 Bin(n, p) 可以用 N(np, np(1 − p)) 来近似。
A common rule is that this approximation is good when np ≥ 5 and n(1 − p) ≥ 5.
常用的判断标准是：np ≥ 5 且 n(1 − p) ≥ 5 时，近似效果较好。
A continuity correction should be applied when converting discreteness to continuity:
在由离散转换为连续时，应使用连续性修正：
P(X = k
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