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Part BCSIR NET December 2025find-a-pivotal-quantity-first-x-over-theta-has-a-theta-free-density-here

Find a pivotal quantity first x over theta has a theta free density here

Let X be a single sample from an absolutely continuous distribution with probability density function f(x∣θ)=(2/θ2)(θ−x)\theta) = (2/\theta^{2})(\theta-x) if 0<x<θ0 < x < \theta, and 0 otherwise, where θ>0\theta > 0 is unknown. Which of the following intervals is a 95% confidence interval for θ\theta?

  1. A.(X,X/(1−0.95))(X, X/(1-\sqrt{0.95}))✓
  2. B.(X, X/0.95)
  3. C.(X,X/0.95)(X, X/\sqrt{0.95})
  4. D.(0.95X, X)

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.

See pricing

50 are analysed free — try those first.

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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