# Seven Deep, Calculational, Predictive Math Problems ## 1) SEIR Epidemic Peak and Capacity Risk - Model: SEIR with N=1,000,000; S(0)=N−60, E(0)=50, I(0)=10, R(0)=0. - Parameters: β=0.30 day⁻¹, σ=1/5 day⁻¹, γ=1/7 day⁻¹; hospitalization rate h=5% of incident infections; length of stay L=10 days; beds B=800. - Rules: - R0 for SEIR: R0 = β/γ. - Peak timing/size via differential equations or next-generation matrix; approximate peak when S/N ≈ 1/R0. - Daily hospital census ≈ Poisson sum of last L days’ admissions; admissions_t ≈ Poisson(h × new_infections_t). - Task: Compute R0, estimate day and size of peak I(t), and estimate the probability that hospital census exceeds B at any time during the outbreak. ## 2) Continuous-Review Inventory: Reorder Point and Backorders - Demand: i.i.d. daily D ~ N(μ=100, σ=40). - Lead time: L=7 days; use (Q,r) with large Q (focus on r). - Service target: Cycle service level CSL=95%. - Rules: - Lead-time demand: D_L ~ N(μL, σ√L). - Reorder point: r = μL + z_CSL σ√L, where z_0.95 ≈ 1.645. - Expected backorders per cycle: E[B] = σ√L φ(z) − z σ√L (1−Φ(z)), z = (r−μL)/(σ√L). - Task: Compute r and the expected total backorders over a 90-day horizon. ## 3) Optimal Risky Portfolio and Shortfall Probability - Assets: - A1: μ1=8%/yr, σ1=20%/yr. - A2: μ2=5%/yr, σ2=10%/yr. - Corr(A1,A2)=0.3; risk-free r_f=2%/yr. - Rules: - Tangency weights (risky-only): w ∝ Σ⁻¹(μ − r_f 1); normalize to sum to 1. - Portfolio mean/variance: μ_p = wᵀμ, σ_p² = wᵀΣw. - Normal return approximation; shortfall: P(R_p < −5%) = Φ((−5% − μ_p)/σ_p). - Task: Find tangency weights for A1,A2, and compute the 1-year probability that return is below −5%. ## 4) Bayesian Defect Prediction and Decision - Prior: p ~ Beta(2,98). - Data: n=1000, x=18 defects. - Next batch: m=500; threshold T=15 defects. - Costs: Accept when true defects rate > threshold implies cost C_A=20; reject when ≤ threshold cost C_R=5 (per batch decision). - Rules: - Posterior: p | data ~ Beta(α+ x, β + n − x). - Posterior predictive: X_next ~ Beta-Binomial(m, α', β'). - Decision: Minimize posterior expected loss; accept if P(X_next > T | data) × C_A ≤ (1 − P(X_next > T | data)) × C_R. - Task: Compute P(X_next > 15 | data) and the Bayes-optimal accept/reject decision. ## 5) Maintenance MDP: Optimal Policy and Cost - States: Good (G), Degraded (D). Failure (F) is an absorbing event that resets to G next week. - Dynamics without maintenance: - G→D with p=0.1/week, stay G otherwise. - D→F with p=0.2/week, stay D otherwise. - Action: “Maintain” in any non-F state resets to G next week; cost c_M=5 when used. Failure incurs c_F=30 and resets to G next week. - Discount: γ=0.95 weekly. - Rules: - Bellman optimality: V(s)=min_a { c(s,a) + γ Σ P(s'|s,a) V(s') }. - Stationary policy π*: compare Q-maintain vs Q-do-nothing in G and D. - Task: Determine π* (maintain in which states), and compute the long-run expected discounted weekly cost under π*. ## 6) Time-Varying Queue Surge: Max Queue and Clearance Time - Queue: M/M/1 with time-varying arrival λ(t) and constant service μ. - Parameters: μ=5/min; λ(t)= - 2/min for 0 ≤ t < 30 min, - 6/min for 30 ≤ t < 90 min, - 2/min for t ≥ 90 min. - Initial condition: empty at t=0. - Rules: - Fluid approximation for expected queue: dQ/dt = λ(t) − μ when λ≠μ, reflecting at 0; piecewise linear in expectation. - Max queue occurs near end of overload interval; clearance time solves Q(t_clear)=0 with λ<μ thereafter. - Task: Using the fluid approximation, compute the expected maximum queue size during the surge and the time after t=90 when the system clears. ## 7) AR(1) Climate Anomaly Exceedance Forecast - Model: X_{t+1} = φ X_t + ε_t, ε_t ~ N(0, σ²), independent; yearly steps. - Parameters: φ=0.90, σ=0.20, current anomaly X_0=1.10 (°C). - Rules: - k-step mean/variance: E[X_k]=φ^k X_0, Var[X_k]=σ² Σ_{i=0}^{k−1} φ^{2i}=σ² (1−φ^{2k})/(1−φ²). - Exceedance probability: P(X_k ≥ τ)=1−Φ((τ−E[X_k])/√Var[X_k]). - Task: For τ=2.0 °C, find the smallest k such that P(X_k ≥ τ) ≥ 10%, and report that k and the corresponding probability.