GATE 2024 ST – Question 50
On Ω={1,2,…}, let P({k})=2^(−k) and X(ω)=ω. Which hold?
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Correct answer: (A) Some k has P(X=k)<10⁻⁶.; (B) lim P(X≥4+1/n)=1/16
Explanation
The masses 2⁻ᵏ tend to zero, proving A. For all n, the threshold 4+1/n selects X≥5 (at n=1 also ≥5), probability Σₖ₌₅∞2⁻ᵏ=1/16, proving B. The interval in C eventually contains no integers, so its probability is zero. Along large even n, D gives P(X≤3)=7/8, while odd n gives P(X≤2)=3/4; no limit exists. **A,B**.