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Part BCSIR NET December 2025the-sum-of-two-uniforms-is-triangular-not-uniform-integrate-the-right-piece

The sum of two uniforms is triangular not uniform integrate the right piece

Let X1X_{1} and X2X_{2} be a random sample from Uniform[0,θ][0,\theta] distribution, where θ>0\theta > 0. For testing the hypothesis H0:θ=1H_{0}: \theta=1 against H1:θ=2H_{1}: \theta=2, consider a test which rejects H0H_{0} if X1+X2>4/5X_{1}+X_{2} > 4/5. Then, the probability of type-I error is

  1. A.8/25
  2. B.13/25
  3. C.17/25✓
  4. D.22/25

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: Any interval of the form [X̄ − (S/√n)t, ∞) with a 90% quantile t is a 90% confidence interval

More on this topic

The chapter behind this: Standard tests and confidence intervals — free to read

From Hypothesis Testing › Likelihood ratio and standard tests

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