GATE 2022 ST – Question 43
X₁,…,X₇ are iid N(0,θ),θ>0. Set K=(X₁²+X₂²)/Σ_(i=1)^7X_i². I:K and ΣX_i² are independent. II:7K/2∼F_(2,7). III:E K²=8/63. Which hold?
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Correct answer: (B) I and III only
Explanation
The numerator/θ and remaining sum/θ are independent χ²₂ and χ²₅. The gamma ratio K is Beta(1,5/2) and independent of their total,proving I. Its second moment is 1·2/[(7/2)(9/2)]=8/63,proving III. II is impossible because 7K/2 is bounded,whereas an F variable is unbounded. **B**.