GATE 2026 ST – Question 42

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

X has support {0,1,2,…} and P(X≥k+1|X≥k)=p for every k≥0,0<p<1. Which holds?

P(X≥k+m|X≥k)=P(X≥m) for every k,m≥0
E(X)>Var(X)
P(X≤x)=1−p^x for integer x≥0
P(X≤k+m|X≥k)=1−p^m for every k,m≥0

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Correct answer: (A) P(X≥k+m|X≥k)=P(X≥m) for every k,m≥0

Explanation

Recursion gives P(X≥k)=p^k. Hence the conditional survival is p^(k+m)/p^k=p^m=P(X≥m), proving memorylessness A. The CDF is 1−p^(x+1), and the last conditional probability is 1−p^(m+1). Mean=p/(1−p), variance=p/(1−p)²>mean. **A**.

GATE 2026 ST – Question 42

Explanation

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