Continuous Random Variables | 连续随机变量

📚 Continuous Random Variables | 连续随机变量

A continuous random variable can take any value within an interval on the real number line. Unlike discrete random variables, which have a probability mass function, continuous random variables are described by a probability density function (PDF). In the IB Mathematics curriculum, understanding continuous random variables is essential for modelling real-world measurements such as time, length, and temperature, where outcomes are uncountably infinite.

连续随机变量可以在实数轴的某个区间上取任意值。与具有概率质量函数的离散随机变量不同，连续随机变量用概率密度函数（PDF）来描述。在IB数学课程中，理解连续随机变量对于建模时间、长度和温度等实际测量至关重要，因为这些结果具有不可数的无限性。

1. Definition and Probability Density Function | 定义与概率密度函数

nction f(x) such that f(x) ≥ 0 for all x, and the total area under the curve is 1, i.e., ∫_-∞^∞ lity that X lies in an interval [a, b] is given by the definite integral: P(a ≤ X ≤ b) = ∫_a^b ous random variable, the probability that X equals any single value is zero: P(X = c) = 0.

连续随机变量 X 具有概率密度函数 f(x)，满足对所有的 x，f(x) ≥ 0，且曲线下的总面积为 1，即 ∫_-∞^∞ f(x) dx = 1。X 落在区间 [a, b] 内的概率由定积分给出：P(a ≤ X ≤ b) = ∫_a^b f(x) dx。注意，对于连续随机变量，X 取任何单一值的概率为零：P(X = c) = 0。

2. Cumulative Distribution Function | 累积分布函数

denoted F(x), gives the probability that X is less than or equal to x: F(x) = P(X ≤ x) = ∫_-∞^x ). Furthermore, where the CDF is differentiable, its derivative recovers the PDF: f(x) = F'(x).

累积分布函数（CDF），记作 F(x)，给出 X 小于或等于 x 的概率：F(x) = P(X ≤ x) = ∫_-∞^x f(t) dt。CDF 是一个非递减函数，且满足 lim_x→-∞ F(x) = 0 和 lim_x→∞ F(x) = 1。对于任意 a < b，P(a ≤ X ≤ b) = F(b) - F(a)。此外，在CDF可导的区间，其导数恢复PDF：f(x) = F'(x)。

3. Expected Value (Mean) | 期望值（均值）

(X) or μ, is the centre of mass of the probability density. It is computed as μ = E(X) = ∫_-∞^∞ absolutely. More generally, the expected value of a function g(X) is given by E(g(X)) = ∫_-∞^∞ g(x) f(x) dx.

连续随机变量 X 的期望值（或均值），记作 E(X) 或 μ，是概率密度的重心。其计算公式为 μ = E(X) = ∫_-∞^∞ x f(x) dx，前提是该积分绝对收敛。更一般地，函数 g(X) 的期望值为 E(g(X)) = ∫_-∞^∞ g(x) f(x) dx。

4. Variance and Standard Deviation | 方差与标准差

f X measures the spread of the distribution around the mean: Var(X) = σ² = E[(X - μ)²] = ∫_-∞^∞ - μ)² f(x) dx. An equivalent computational formula is Var(X) = E(X²) - μ², where E(X²) = ∫_-∞^∞ f(x) dx. The standard deviation is the positive square root of the variance: σ = √Var(X).

X 的方差衡量分布围绕均值的离散程度：Var(X) = σ² = E[(X - μ)²] = ∫_-∞^∞ (x - μ)² f(x) dx。一个等价的计算公式为 Var(X) = E(X²) - μ²，其中 E(X²) = ∫_-∞^∞ x² f(x) dx。标准差是方差的正平方根：σ = √Var(X)。

5. Mode of a Continuous Distribution | 连续分布的众数

may be unimodal (one peak) or multimodal (several local maxima). The mode is not necessarily unique, and it can be found by solving f'(x) = 0 and checking the second derivative or by inspecting the shape of the density.

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连续随机变量的众数是使概率密度函数 f(x) 取得全局最大值的 x 值。分布可以是单峰的（一个峰值）或多峰的（多个局部最大值）。众数不一定唯一，可以通过解 f'(x) = 0 并检查二阶导数，或通过观察密度形状来求得。

6. Median and Percentiles | 中位数与百分位数

s: P(X ≤ m) = F(m) = 0.5. More generally, the 100p-th percentile (0 < p < 1) satisfies F(x_p quartile Q₃ to p = 0.75. These measures are often found by solving the integral equation ∫_-∞^x_p f(t) dt = p.

中位数 m 是将概率分布分成两个相等部分的值：P(X ≤ m) = F(m) = 0.5。更一般地，第 100p 百分位数（0 < p < 1）满足 F(x_p) = p。例如，下四分位数 Q₁ 对应 p = 0.25，上四分位数 Q₃ 对应 p = 0.75。这些度量通常通过求解积分方程 ∫_-∞^x_p f(t) dt = p 来求得。

7. Linear Transformations | 线性变换

Var(Y) = a² Var(X). The probability density function of Y can be derived from that of X: f_Y(y) = (1/|a|) f_X((y – b)/a). The cumulative distribution function transforms as F_Y(y) = F_X((y – b)/a) for a > 0, with a sign reversal if a < 0.

若 X 是连续随机变量，且 Y = aX + b 其中 a ≠ 0，则 Y 的期望和方差分别为 E(Y) = aE(X) + b 和 Var(Y) = a² Var(X)。Y 的概率密度函数可由 X 的密度函数导出：f_Y(y) = (1/|a|) f_X((y – b)/a)。累积分布函数的变换为：当 a > 0 时 F_Y(y) = F_X((y – b)/a)，若 a < 0 则需符号调整。

8. Standardization and the Normal Distribution | 标准化与正态分布

continuous distributions is the normal distribution with mean μ and variance σ², written X ~ N(μ, σ²). Its density is f(x) = (1/(σ√2π)) e ^{–(x – μ)²/(2σ²)} . By standardizing, Z = (X – μ)/σ, we obtain the standard normal distribution Z ~ widely tabulated. Probabilities for any normal variable can be found using P(X ≤ a) = Φ((a – μ)/σ).

最重要的连续分布之一是均值为 μ、方差为 σ² 的正态分布，记作 X ~ N(μ, σ²)。其密度函数为 f(x) = (1/(σ√2π)) e ^{–(x – μ)²/(2σ²)} 。通过标准化 Z = (X – μ)/σ，我们得到标准正态分布 Z ~ N(0,1)，其 CDF Φ(z) 被广泛列于表格中。任意正态变量的概率可通过 P(X ≤ a) = Φ((a – μ)/σ) 计算。

9. Expectation and Variance of Sums of Independent Variables | 独立变量之和的期望与方差

ies are fundamental for combining distributions and appear frequently in IB exam questions on linear combinations of random variables.

对于两个连续随机变量 X 和 Y，线性组合的期望满足 E(aX + bY) = aE(X) + bE(Y)，无论它们是否独立。若 X 与 Y 独立，则方差满足 Var(aX + bY) = a² Var(X) + b² Var(Y)。这些性质是组合分布的基础，在IB考试中关于随机变量线性组合的问题里频繁出现。

10. The Continuous Uniform Distribution | 连续均匀分布

tinuous uniform random variable X on the interval [α, β] has constant density: f(x) = 1/(β – α) for α ≤ x ≤ β, and zero elsewhere. Its CDF is F(x) = (x – α)/(β – α) for α ≤ x ≤ β. The mean is E(X) = (α + β)/2, and the variance is Var(X) = (β – ions where all values in an interval are equally likely, such as random number generation.

在区间 [α, β] 上的连续均匀随机变量 X 具有常数密度：当 α ≤ x ≤ β 时，f(x) = 1/(β – α)，其他处为零。其 CDF 为 F(x) = (x – α)/(β – α)（α ≤ x ≤ β）。均值为 E(X) = (α + β)/2，方差为 Var(X) = (β – α)²/12。该分布用于建模区间内所有值等可能的情形，例如随机数生成。

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