A network engineer models the inter-arrival time T (in microseconds) between packets as an exponential random variable with rate λ = 500 000 μs⁻¹. The resulting probability density function is
f_T(t) = 500 000 e^(−500 000 t), t ≥ 0.
Seeing that f_T(0) = 500 000 ≫ 1, the engineer worries that the model violates the fundamental rule that probabilities cannot exceed 1. Which explanation best resolves the engineer's concern?
Any PDF whose maximum exceeds 1 indicates incorrect parameter estimation; λ must be reduced until the entire density is ≤ 1.
The apparent problem arises from using microseconds; dividing the density by 1 000 000 to convert it to seconds would ensure it never exceeds 1.
The value 500 000 is acceptable because a PDF expresses probability per microsecond; actual probabilities are obtained by integrating over an interval, and the total area still equals 1.
The value at t = 0 is ignored since the exponential distribution is undefined at zero; therefore 500 000 has no bearing on the model's validity.
A probability density function reports probability per unit of measurement, not probability itself. Although the exponential PDF attains the value 500 000 at t = 0 μs, the probability of observing T in any finite interval is obtained by integrating the density over that interval. Because the area under the entire curve ∫₀∞ 500 000 e(−500 000 t) dt equals 1, no probability rule is violated. Simply changing units or the rate parameter can alter the numerical value of the density at a point, but it will still be valid as long as the total integrated area remains 1. The other options are incorrect because the PDF is well-defined at t = 0, scaling units alone does not 'fix' an error, and there is no requirement that a PDF stay below 1 everywhere.
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Why can the PDF exceed 1 at a specific point without violating probability rules?
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What does it mean to integrate a PDF, and why is this important?
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Why doesn't scaling the unit of measurement (e.g., switching from microseconds to seconds) fix the concern?