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Part CCSIR NET December 2025with-alpha-and-beta-unspecified-the-prior-mean-is-a-free-parameter-no-directional-claim-survives-that

With alpha and beta unspecified the prior mean is a free parameter no directional claim survives that

Suppose X∣θ\theta ~ Binomial(7,θ),0<θ<1(7,\theta), 0<\theta<1, and the prior distribution of θ\theta is Beta(α,β)(\alpha,\beta) where α>0\alpha>0 and β>0\beta>0 are known. Then which of the following statements MAY NOT be true?

  1. A.Posterior mean of θ\theta given X=2 is less than the prior mean✓
  2. B.Posterior mean of θ\theta given X=3 is greater than the prior mean✓
  3. C.Posterior mean of θ\theta given X=4 is not equal to the prior mean✓
  4. D.Posterior mean of θ\theta given X=5 is equal to the prior mean✓

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.

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50 are analysed free — try those first.

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

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