📚 IB Mathematics: Continuous Random Variables Key Points | IB数学：连续随机变量考点精讲
A continuous random variable can take any real value in an interval or union of intervals. Unlike discrete random variables, probabilities are not assigned to single points but to intervals, and the main tool is the probability density function (pdf). This article reviews the essential concepts and exam-style skills for IB Mathematics.
连续随机变量可以在一个区间或若干区间的并集内取任意实数值。与离散随机变量不同，连续随机变量不是把概率分配给单个点，而是分配给区间；核心工具是概率密度函数（pdf）。本文系统梳理IB数学中连续随机变量的必备考点与解题技能。
1. Continuous Random Variables and Probability Density Function | 连续随机变量与概率密度函数
A continuous random variable X is described by a probability density function f(x). The pdf is not a probability itself; it gives the density of probability at each point. For any interval [a, b], the probability is the area under the graph of f(x) from a to b.
连续随机变量 X 由概率密度函数 f(x) 描述。f(x) 本身不是概率，而是表示每个点附近的概率密度。对任意区间 [a, b]，概率等于 f(x) 图像从 a 到 b 下方的面积。
- The pdf is defined for all real x, but often has a restricted domain.
- For a single point x = c, P(X = c) = 0.
- 概率密度函数对所有实数 x 有定义，但通常只在某个范围内取非零值。
- 对单个点 x = c，有 P(X = c) = 0。
2. Properties of a Valid pdf | 概率密度函数的合法条件
For f(x) to be a valid probability density function, two conditions must be satisfied:
要使 f(x) 成为合法的概率密度函数，必须满足两个条件：
f(x) ≥ 0 for all x, and ∫−∞∞ f(x) dx = 1
If the support is [a, b], the second condition becomes ∫ab f(x) dx = 1.
如果定义域为 [a, b]，第二个条件变为 ∫ab f(x) dx = 1。
- Do not forget to check f(x) ≥ 0 inside the domain.
- The total area under any valid pdf must equal exactly 1.
- 不要忘记检查定义域内 f(x) ≥ 0。
- 任何合法 pdf 曲线下的总面积必须正好等于 1。
3. Cumulative Distribution Function | 累积分布函数
The cumulative distribution function (cdf) is defined as F(x) = P(X ≤ x) = ∫−∞x f(t) dt. It accumulates probability from the left up to x.
累积分布函数（cdf）定义为 F(x) = P(X ≤ x) = ∫−∞x f(t) dt，它从最左侧累积到 x 处的概率。
- F(x) is non-decreasing as x increases.
- F(−∞) = 0 and F(∞) = 1.
- If F is known, the pdf is f(x) = F′(x) wherever differentiable.
- F(x) 随 x 增大而不减。
- F(−∞) = 0，F(∞) = 1。
- 若已知 F，则在可导处 pdf 为 f(x) = F′(x)。
4. Calculating Probabilities as Areas | 用面积计算概率
For a continuous random variable, P(a < X < b) = ∫ab Since single points have probability zero, open and closed endpoints give the same result.
对连续随机变量，P(a < X < b) = ∫ab f(x) dx = F(b) − F(a)。由于单点的概率为零，开区间与闭区间的结果相同。
- Always integrate the pdf over the required interval, not over the whole domain.
- If the pdf is piecewise, split the integral at the breakpoints.
- Use GDC or symmetry when the integral has no simple antiderivative.
- 始终对所需区间上的 pdf 积分，而不是在整个定义域上积分。
- 如果 pdf 是分段函数，要在分段点处拆分积分。
- 当积分没有初等原函数时，可用计算器或利用对称性。
5. Expected Value | 期望值
The expected value (mean) of a continuous random variable is the long-run average value of X. It is calculated by integrating x times the pdf:
连续随机变量的期望值（均值）是 X 的长期平均值，计算公式为对 x 乘以 pdf 求积分：
E(X) = μ = ∫−∞∞ x f(x) dx
More generally, E(g(X)) = ∫−∞∞ g(x) f(x) dx.
更一般地，E(g(X)) = ∫−∞∞ g(x) f(x) dx。
- For X ≥ 0, use ∫0∞ x f(x) dx.
- If the integral converges, it gives the mean; otherwise the mean does not exist.
- 若 X ≥ 0，则使用 ∫0∞ x f(x) dx。
- 若积分收敛则得到均值；若发散则均值不存在。
6. Variance and Standard Deviation | 方差与标准差
Variance measures the spread of the distribution around the mean. The formula is:
方差衡量分布围绕均值的离散程度，公式为：
Var(X) = E(X²) − [E(X)]² = ∫−∞∞ x² f(x) dx − μ²
The standard deviation is σ = √Var(X).
标准差为 σ = √Var(X)。
- Always compute E(X²) separately from [E(X)]².
- Use GDC to check indefinite integration when f(x) is complicated.
- For linear transformations: Var(aX + b) = a² Var(X).
- 务必分别计算 E(X²) 与 [E(X)]²，不要混淆。
- 当 f(x) 复杂时，可用计算器检验积分结果。
- 线性变换下：Var(aX + b) = a² Var(X)。
7. Mode, Median and Quantiles | 众数、中位数与分位数
the support. The median m satisfies F(m) = 0.5, meaning half the probability lies below m.
众数是 pdf f(x) 在定义域上取得最大值的 x 值。中位数 m 满足 F(m) = 0.5，即一半概率位于 m 以下。
The p-quantile qₚ is the value such that P(X ≤ qₚ) = p, or ∫−∞qₚ f(x) dx = p.
p 分位数 qₚ 满足 P(X ≤ qₚ) = p，即 ∫−∞qₚ f(x) dx = p。
- For a symmetric distribution, mean = median = mode.
- To find median, solve F(x) = 0.5, not E(X) = 0.5.
- Quantiles are often found by setting the cdf equal to p and solving.
- 对对称分布，均值 = 中位数 = 众数。
- 求中位数是解 F(x) = 0.5，而不是 E(X) = 0.5。
- 分位数通常令 cdf 等于 p 再解方程。
8. Uniform Distribution | 均匀分布
If X follows a uniform distribution on [a, b], written X ~ U(a, b), its pdf is constant over the interval:
若 X 服从 [a, b] 上的均匀分布，记作 X ~ U(a, b)，其 pdf 在区间上为常数：
f(x) = 1 / (b − a), for a ≤ x ≤ b; otherwise 0
- E(X) = (a + b) / 2
- Var(X) = (b − a)² / 12
- CDF: F(x) = (x − a) / (b − a) for a ≤ x ≤ b
- E(X) = (a + b) / 2
- Var(X) = (b − a)² / 12
- CDF：F(x) = (x − a) / (b − a)，a ≤ x ≤ b
9. Exponential Distribution and Memorylessness | 指数分布与无记忆性
An exponential random variable X ~ Exp(λ) models waiting time. Its pdf and cdf are:
指数随机变量 X ~ Exp(λ) 常用于建模等待时间，其 pdf 与 cdf 为：
f(x) = λ exp(−λx) for x ≥ 0, F(x) = 1 − exp(−λx)
- E(X) = 1 / λ, Var(X) = 1 / λ².
- Memoryless property: P(X > s + t | X > s) = P(X > t).
- This means the remaining waiting time is independent of how long you have already waited.
- E(X) = 1 / λ，Var(X) = 1 / λ²。
- 无记忆性：P(X > s + t | X > s) = P(X > t)。
- 这意味着剩余等待时间与已经等待了多久无关。
10. Normal Distribution and Standardisation | 正态分布与标准化
If X ~ aped curve. Probabilities are found using the standard normal distribution Z = (X − μ) / σ ~ N(0, 1).
若 X ~ N(μ, σ²)，其 pdf 是经典钟形曲线。概率可通过标准正态分布 Z = (X − μ) / σ ~ N(0, 1) 计算。
f(x) = 1 / (σ √2π) exp(−(x − μ)² / (2σ²))
- Standardise before using the normal table or GDC.
- Use symmetry: P(Z < −z) = P(Z > z) = 1 − Φ(z).
- For a normal distribution, the total area under the curve is still 1.
- 使用正态分布表或计算器前先标准化。
- 利用对称性：P(Z < −z) = P(Z > z) = 1 − Φ(z)。
- 正态分布曲线下的总面积为 1。
11. Inverse Normal Calculations | 正态分布逆运算
In IB problems you may be given a probability and asked to find the value of μ, σ, or a quantile. This requires using the inverse normal function on a GDC, often written as invNorm(p, μ, σ).
IB 考试中有时会给概率，要求反过来求 μ、σ 或分位数。这时需要使用计算器上的逆正态函数，通常写为 invNorm(p, μ, σ)。
- If P(X < k) = p, then k = invNorm(p, μ, σ).
- If P(X > k) = p, first consider 1 − p and then use invNorm.
- Be careful with one-sided and two-sided intervals; draw a diagram.
- 若 P(X < k) = p，则 k = invNorm(p, μ, σ)。
- 若 P(X > k) = p，先用 1 − p 再调用 invNorm。
- 注意单侧区间与双侧区间的区别，务必画图辅助。
12. Key Points and Common Mistakes | 考点总结与常见错误
Continuous random variables appear frequently in IB Paper 2 and Paper 3. The most reliable approach is: identify the support, check that the pdf is valid, then integrate or differentiate correctly.
连续随机变量在 IB Paper 2 和 Paper 3 中经常出现。最稳妥的步骤是：确定定义域、验证 pdf 合法性，然后正确地进行积分或求导。
- Remember P(X = a) = 0 for continuous X.
- Do not use the cdf value directly as a probability at a point.
- Check whether the distribution is uniform, exponential, or normal; each has its own formulas.
- Always quote units if the variable has units.
- 记住连续随机变量有 P(X = a) = 0。
- 不要把 cdf 在某点的值当作“单点概率”。
- 判断分布是均匀、指数还是正态，并选用相应公式。
- 若变量有单位，答案中要写清单位。
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