The iid X_n have density e^(−(x−θ)),x≥θ,zero otherwise,θ>0. I:the sample average converges in probability to (θ+1)/2. II:lim E[min(X₁,…,X_n)]=θ. Which hold?
X_i=θ+E_i with iid rate-one exponentials, so its mean is θ+1 and the law of large numbers refutes I. The minimum is θ plus an exponential of rate n, expectation θ+1/n→θ. Only II, **B**.
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