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GATE 2021 ST – Question 41

Standard Univariate Distributions · Discrete Distributions · 2 marks · Multiple choice

X is Poisson with E X²=110. Which claim is NOT true?

  1. E X^n=10E(X+1)^(n−1) for every integer n≥1.
  2. P(X even)=(1+e⁻²⁰)/4.
  3. P(X=k)<P(X=k+1) for k=0,…,8.
  4. P(X=k)>P(X=k+1) for k=10,11,… .

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Show answer and explanation

Correct answer: (B) P(X even)=(1+e⁻²⁰)/4.

Explanation

The mean λ solves λ²+λ=110,so λ=10. The Poisson shift identity proves A. Consecutive probabilities have ratio 10/(k+1),giving C,D. Parity follows from E(−1)^X=e⁻²⁰:the even probability is (1+e⁻²⁰)/2,not divided by 4. **B**.

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