MLE for mean lifetime of LED bulbs

Question

The operating lifetime of an LED bulb, in thousands of hours, is modeled as exponential with unknown mean $\theta$ , density $f(x \mid \theta) = \frac{1}{\theta} e^{-x/\theta}$ for $x > 0$ . Five bulbs are run to failure at lifetimes 9, 12, 6, 15, 8.

Derive and compute the maximum likelihood estimate of the mean lifetime $\theta$ .

Accepted Answer

For n independent exponential observations the log-likelihood is \ell(	heta) = -n\ln	heta - \frac{1}{	heta}\sumi xi. Differentiating and solving \ell'(	heta) = 0 yields the sample mean: \hat	heta = \frac{1}{n}\sumi xi = \bar x. With 9+12+6+15+8 = 50 and n = 5, \hat	heta = \frac{50}{5} = 10 	ext{ (thousand hours)}. Takeaway. For exponential lifetimes the maximum likelihood mean is simply the average observed lifetime.

MLE for mean lifetime of LED bulbs

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