Q11Multiple correct3 Marks13 Apr 2025Suppose X1,X2,…,Xn∼i.i.d. Uniform(0,θ)X_1, X_2, \dots, X_n \sim \text{i.i.d. Uniform}(0, \theta)X1,X2,…,Xn∼i.i.d. Uniform(0,θ), and let θ^=kn∑i=1nXi\hat{\theta} = \frac{k}{n}\sum_{i=1}^n X_iθ^=nk∑i=1nXi be an estimator of θ\thetaθ. Which of the following option(s) is(are) correct?AE[θ^]=k⋅n(n+1)2E[X1]E[\hat{\theta}] = k \cdot \frac{n(n+1)}{2} E[X_1]E[θ^]=k⋅2n(n+1)E[X1]Bθ^\hat{\theta}θ^ is an unbiased estimator of θ\thetaθ if and only if k=2k = 2k=2.CVar(θ^)=k2θ212n\text{Var}(\hat{\theta}) = \frac{k^2 \theta^2}{12 n}Var(θ^)=12nk2θ2DRisk=θ23n\text{Risk} = \frac{\theta^2}{3n}Risk=3nθ2 for k=2k = 2k=2
