Problem 6.1: worked solution
The analytic transmission at tau=2 is exp(-2)=0.1353353. The nested seeded sample gives:
| N | successes (reconstructed from p_hat*N) | p_hat | plug-in SE | |p_hat-p| |
|---|---|---|---|---|
| 100 | 12 | 0.12000 | 0.03250 | 0.01534 |
| 500 | 65 | 0.13000 | 0.01504 | 0.00534 |
| 2000 | 271 | 0.13550 | 0.00765 | 0.00016 |
| 20000 | 2756 | 0.13780 | 0.00244 | 0.00246 |
The nominal SE is sqrt(p_hat(1-p_hat)/N); an interval such as p_hat +/- 1 SE is a diagnostic, not a universal confidence statement, especially at small N. Prefix estimates are correlated. Repeat with independent seeds for empirical coverage. Exact binomial intervals are an acceptable alternative and are preferable near probability boundaries.
Problem 6.2: worked solution
Let the target success probability be p=exp(-4)=0.01831564 and proposal success probability q=0.2. The target/proposal likelihood ratio is p/q=0.09157819 on success and (1-p)/(1-q) on failure; for the indicator f=1(success), the failure contribution is zero. Therefore E_q[1(success)*p/q]=q*p/q=p, proving unbiasedness. At N=10,000, ordinary sampling gives 0.01920000 +/- 0.00137234; weighted sampling gives 0.01814164 +/- 0.00036502 (one estimated SE). The result is seed-specific; compare multiple independent runs or analytic estimator variance. A derivation using sums over the two outcomes is equivalent.
Problem 6.3: worked solution
For v(t)=gamma+K sin(2*pi*t/11.2+phi), the seeded weighted fit gives K=20.8286 m/s, phi=0.69746 rad, gamma=3.1325 m/s; local standard errors are 0.5603 m/s, 0.02683 rad, and 0.3957 m/s under the source's Jacobian covariance with residual-scale estimate. Weighted chi-square=67.882 for 61 degrees of freedom; residual RMS=3.0897 m/s. Inspect residuals versus time and phase: a coherent additional periodicity, long-term trend, or phase-dependent variance would invalidate the fixed-period one-sinusoid/noise model. Because the supplied sigma is known, covariance without reduced-chi-square rescaling is also acceptable if explicitly justified; a linear sin/cos fit is an independent parameterization.
Scoring guide
Each problem is scored for model/setup (25%), correct computation or derivation (30%), validation and uncertainty (25%), and physical interpretation/communication (20%). Accept alternative correct methods when they meet the same evidence standard. See assessment instructions for error classification and resubmission guidance.