The GATE Grind

GATE 2026 ST – Question 55

Stochastic Processes · Poisson and Birth-Death Processes · 2 marks · Multiple select

N(t) is a homogeneous Poisson process of rate 3 and T₁ its first arrival time. Which hold?

  1. E(T₁)=3
  2. E(T₁)=1/3
  3. E(T₁|N(2)=4)=6/5
  4. E(T₁|N(3)=4)=3/5

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: (B) E(T₁)=1/3; (D) E(T₁|N(3)=4)=3/5

Explanation

T₁ is exponential of rate 3, mean 1/3. Conditional on N(t)=4, the four arrival times are the ordered values of four iid uniforms on (0,t), whose minimum has mean t/5. Thus the conditional means are 2/5 and 3/5 respectively. **B,D**.

More Poisson and Birth-Death Processes questions

All Poisson and Birth-Death Processes questions →