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Part CCSIR NET December 2024overlapping-index-ranges-are-not-independent

Overlapping index ranges are not independent

Let X1,X2,…,X13X_{1}, X_{2}, \dots, X_{13} be independent and identically distributed (i.i.d.) Poisson(θ)(\theta) random variables, where θ>0\theta > 0. Then, which of the following statements are true?

  1. A.(∑(i=1..8)Xi)(∑(i=8..13)Xi)(\sum(i=1..8)X_{i})(\sum(i=8..13)X_{i}) is an unbiased estimator of 48θ248\theta^{2}.
  2. B.(1/13)∑(i=1..13)Xi(1/13)\sum(i=1..13)X_{i} is method of moments estimator of θ\theta.✓ Correct
  3. C.There does not exist any unbiased estimator of e(−7θ)e^(-7\theta).
  4. D.(e(X7),∑(i=1..6)Xi,∑(i=8..13)Xi)(e^(X_{7}), \sum(i=1..6)X_{i}, \sum(i=8..13)X_{i}) is a sufficient statistic for θ\theta.✓ Correct

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward, and why this one tests dependence misread.

See pricing

50 are analysed free — try those first.

The trap it tests

Dependence misread

Independence, pairwise vs mutual, or correlation vs dependence.

Drill statements like this

Related counterexample: A sufficient statistic is complete

More on this topic

The chapter behind this: Sufficiency, completeness and the UMVUE machine — free to read

From Estimation › Sufficiency, completeness, UMVUE, Cramér–Rao

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