GATE 2024 ST – Question 62
A Poisson process N(t) of unknown rate λ counts road potholes up to distance t km. Starting at zero, potholes occur at distances .9,1.3,1.8,2.7,3.4,4.1,4.7,5.5,6.2,6.8,7.4,8.1,8.9,9.2,9.7 km. Find the method-of-moments estimate of λ to two decimals.
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Correct answer: 0.60 to 0.70
Explanation
The official accepted interval is **.60–.70**, matching the mean of 15 interarrival gaps: total distance 9.7 km divided by 15 is **.65 km per pothole**. There is a rate/scale discrepancy in the printed question and key. For a Poisson rate λ measured in potholes per km, gaps have mean 1/λ; moment matching gives λ̂=15/9.7≈**1.55 potholes/km**, the reciprocal. The official grading is retained, and the solution states explicitly that its keyed .65 is the estimated mean gap rather than the stated rate.