The GATE Grind

GATE 2024 ST – Question 22

Stochastic Processes · Brownian Motion · 1 mark · Multiple choice

For standard Brownian motion W, which is not true?

  1. E W(7)=0
  2. E[W(5)W(9)]=7
  3. 2W(1)~N(0,4)
  4. E[W(5)|W(3)=3]=3

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Correct answer: (B) E[W(5)W(9)]=7

Explanation

The covariance formula gives E[W(5)W(9)]=min(5,9)=5, not 7. Brownian motion has mean zero, variance at time 1 equal to 1, and its future increment after time 3 is independent and mean zero, verifying the others. **B**.

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