GATE 2024 ST – Question 23
Independent X₁,X₂,X₃ are Binomial(100,p),0<p<1 unknown. Let T₁=(X₁+X₂,X₃),T₂=X₁+X₂+X₃. I:T₂|T₁ has a parameter-free distribution. II:T₁|T₂ has a parameter-free distribution. Which hold?
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Correct answer: (C) Both I and II
Explanation
T₂ is a fixed function of T₁, so the first conditional law is a point mass independent of p. T₂ is sufficient for p in the product-binomial likelihood; given total t, the allocation X₃ is hypergeometric with population sizes 100 and 200 among 300 trials, not involving p. This determines T₁ and proves II. **C**.