概率论与数理统计（占总分40%–50%）
该模块是复旦应用统计考研的基石内容，题目覆盖全面且注重概念辨析与推导严谨性。常见题型包括：
- 概率公理系统应用：如利用贝叶斯公式反推后验概率，常结合医学诊断或工业质检场景设计条件概率陷阱
- 极限定理综合运用：2021年真题考查“用林德伯格-莱维中心极限定理近似计算二项分布概率”，要求先验证独立同分布条件，再进行连续性修正
- 充分性与完备性证明：如证明指数族分布中样本均值为充分完备统计量，并据此构造UMVUE
- 区间估计与假设检验的等价性：要求通过置信区间反推检验拒绝域，或反之
【典型例题】2023年计算题：设X₁,…,Xₙ iid ~ Uniform(0,θ)，θ>0。构造θ的置信水平为1−α的置信区间，并证明该区间长度的期望最小。
★ 解题关键：利用极小充分统计量Y=max{Xᵢ}的分布（f(y)=nyⁿ⁻¹/θⁿ），构造 pivotal quantity Y/θ，再通过P(a≤Y/θ≤b)=1−α确定a,b使区间长度b−a最小（需解优化问题：min(b−a) s.t. bⁿ−aⁿ=1−α）
随机过程（占总分10%–20%）
复旦真题中的随机过程题目虽占比不高，但难度陡增，常与实际问题结合考查建模能力。重点包括：
- 马尔可夫链的转移概率与平稳分布：要求计算有限状态不可约链的平稳分布π（解线性方程组πP=π），并判断其唯一性与遍历性
- 泊松过程的复合与分解：如“顾客到达服从泊松过程，每位顾客购买商品数独立同分布，求单位时间总销售额的分布”
- 维纳过程与伊藤积分基础：2020年真题考查“应用伊藤引理求d(Wₜ³)”
- 生灭过程的应用建模：如排队系统、疾病传播模型的转移速率矩阵构建
【典型例题】设某服务台顾客到达为强度λ的泊松过程，服务时间服从参数μ的指数分布。若系统容量有限为N（含正在服务者），求系统稳态损失概率。
★ 解题路径：构建M/M/1/N排队模型的状态空间{0,1,…,N}，写出转移速率矩阵Q，求解平衡方程πQ=0，得πₙ = 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