The GATE Grind

GATE 2021 ST – Question 48

Joint Distributions · Order Statistics · 2 marks · Multiple choice

X_(1)<⋯<X_(5) are order statistics of a size-five Uniform(0,θ) sample. P:3X_(2) is unbiased for θ. Q:Var[E(2X_(3) | X_(5))]≤Var(2X_(3)). Which hold?

  1. P only
  2. Q only
  3. Both P and Q
  4. Neither P nor Q

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: (C) Both P and Q

Explanation

The uniform order mean is E X_(k)=kθ/(n+1). Hence E(3X_(2))=3·2θ/6=θ. For Q,the law of total variance gives Var(2X_(3))=Var[E(2X_(3)|X_(5))]+E Var(2X_(3)|X_(5)),and the last term is nonnegative. Both hold. **C**.

More Order Statistics questions

All Order Statistics questions →