S² is the sample variance with divisor n−1 from n>1 iid normal observations with unknown μ and σ²>0. I:S² is unbiased for σ² and S is unbiased for σ. II:((n−1)/n)S² is the MLE of σ² and √((n−1)/n)S is the MLE of σ. Which hold?
S² is unbiased,but strict concavity of the square root gives E S<√(E S²)=σ,so I is false. Normal likelihood gives the variance MLE Σ(X_i−X̄)²/n=((n−1)/n)S². Invariance gives its positive square root as the σ MLE. Thus **B**.
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