Skip to content
Part BCSIR NET June 2025a-flat-improper-prior-adds-no-information-so-the-posterior-variance-does-not-shrink

A flat improper prior adds no information so the posterior variance does not shrink

Suppose the distribution of X given θ\theta is normal with mean θ\theta and variance 15. Further, let the prior (improper) distribution of θ\theta be proportional to 1,−∞<θ<∞1, -\infty < \theta < \infty. If the observed value of X is 13, then which of the following statements is true?

  1. A.Posterior mean = Maximum likelihood estimate of θ\theta, Posterior variance = Var(X|θ)\theta)✓ Correct
  2. B.Posterior mean = Maximum likelihood estimate of θ\theta, Posterior variance < Var(X|θ)\theta)
  3. C.Posterior mean > Maximum likelihood estimate of θ\theta, Posterior variance = Var(X|θ)\theta)
  4. D.Posterior mean > Maximum likelihood estimate of θ\theta, Posterior variance < Var(X|θ)\theta)

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 hypothesis dropped.

See pricing

50 are analysed free — try those first.

The trap it tests

Hypothesis dropped

A theorem applied without checking one of its preconditions.

Drill statements like this

Related counterexample: Pairwise independent events are mutually independent

More on this topic

The chapter behind this: Probability axioms, conditioning and independence — free to read

From Probability › Axioms, conditional probability, independence, Bayes

Last revised . Found a mistake? Tell us — corrections are the fastest thing we act on.

ShareWhatsAppTelegram