GATE 2023 ST – Question 54

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

X has PMF P(X=0)=(1+e⁻¹)/2 and P(X=k)=e⁻¹/(2k!) for k≥1. Which hold?

E X=1
E X<1
E(X | X>0)<1/2
E(X | X>0)>1/2

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Show answer and explanation

Correct answer: (B) E X<1; (D) E(X | X>0)>1/2

Explanation

This is a 50:50 mixture of a point mass at zero and Poisson(1). Thus E X=1/2<1. Also P(X>0)=(1−e⁻¹)/2,so conditional mean is (1/2)/[(1−e⁻¹)/2]=1/(1−e⁻¹)>1/2. **B,D**.

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