Gamma scale MLE for waiting times with shape five

Question

The total time to accumulate five independent radioactive counts is Gamma with known shape $k = 5$ and unknown scale $\theta$ , density $f(x \mid \theta) = \dfrac{x^{k-1} e^{-x/\theta}}{\Gamma(k)\,\theta^{k}}$ for $x > 0$ . Repeating the experiment many times yields a sample mean of $\bar x = 15$ seconds.

Estimate the scale $\theta$ by maximum likelihood.

Accepted Answer

The 	heta-dependent part of the log-likelihood is -nk\ln	heta - \frac{1}{	heta}\sumi xi, and the stationary point is \hat	heta = \frac{\bar x}{k} = \frac{15}{5} = 3 	ext{ seconds}. As a sanity check, the Gamma mean k	heta = 5 	imes 3 = 15 matches \bar x, exactly what maximum likelihood enforces. Takeaway. The scale MLE makes the model mean k	heta equal the sample mean, so \hat	heta = \bar x / k.

Gamma scale MLE for waiting times with shape five

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