MLE of exponential mean from summary totals

Question

Latencies for a network service are modeled as independent exponential variables with unknown mean $\theta$ , density $f(x \mid \theta) = \frac{1}{\theta} e^{-x/\theta}$ for $x > 0$ . You are told only that $n = 20$ latencies were recorded and that their sum was 150 milliseconds; the individual values were not kept.

What is the maximum likelihood estimate of $\theta$ ?

Accepted Answer

The MLE of an exponential mean depends on the data only through the total, because the log-likelihood is \ell(	heta) = -n\ln	heta - \frac{1}{	heta}\sumi xi, and solving \ell'(	heta) = 0 gives \hat	heta = \frac{1}{n}\sumi xi. The sum \sumi xi is a sufficient statistic, so knowing the total and the count is all you need: \hat	heta = \frac{150}{20} = 7.5 	ext{ milliseconds}. Takeaway. The exponential MLE uses only the sum and the count, so summary totals are enough to compute it.

MLE of exponential mean from summary totals

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