MLE for mean time between support tickets

Question

A help desk models the time in minutes between arriving support tickets as independent exponential variables with unknown mean $\theta$ , density $f(x \mid \theta) = \frac{1}{\theta} e^{-x/\theta}$ for $x > 0$ . Four gaps are observed: 10, 4, 7, 3 minutes.

Find the maximum likelihood estimator of $\theta$ and evaluate it here.

Accepted Answer

The log-likelihood for n independent exponential observations is \ell(	heta) = -n\ln	heta - \frac{1}{	heta}\sumi xi. Setting \ell'(	heta) = -\dfrac{n}{	heta} + \dfrac{1}{	heta^2}\sumi xi = 0 gives \hat	heta = \bar x, the sample mean. Here 10+4+7+3 = 24 over n = 4, so \hat	heta = \frac{24}{4} = 6 	ext{ minutes}. Takeaway. Whatever the setting, the exponential mean parameter is estimated by the plain average of the observed waits.

MLE for mean time between support tickets

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