GATE 2024 ST – Question 54
An iid Poisson(λ) sample n≥2 defines T₁=X̄ and T₂=√[Σ(X_i−X̄)²/(n−1)]. Which hold?
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Correct answer: (A) T₁ is unbiased for λ.; (C) T₂² is unbiased for λ.; (D) Both T₁ and T₂ are estimators of λ as well as λ².
Explanation
The mean has expectation λ, and the usual unbiased sample variance T₂² has expectation Var X=λ, proving A,C. Strict concavity of the square root makes E T₂<√λ, so B fails. The official key also includes D in the broad formal meaning of “estimator”: any statistic can be proposed to estimate either target without a claim of unbiasedness, consistency or quality. D should not be read as asserting those stronger properties. **A,C,D**.