As we saw in Exercise 40.19, the half-life of a radioactive substance is the time taken for half of its atoms to decay. If atoms decay according to an exponential distribution with rate

λ per unit time, then the half-life can be calculated from t1/2 =

− ln(0.5)

λ

.

In this question we’ll use our methodology for estimating λ to find an estimator for the half-life, t1/2. We’ll also investigate the properties of our estimator, and work out how to calculate a confidence interval.
Suppose we observe k atomic decay times. The mean of the k observations is X.

a. What is the maximum likelihood estimator of the decay rate, λ?

b. Using your answer to (a), suggest a suitable estimator for the half-life. Call your estimatorbt1/2.

c. Find the mean and variance of your estimator bt1/2. Is it unbiased?
Since λb is an MLE, your estimator bt1/2 is also an MLE. This is because we could reparametrise the likelihood in terms of t1/2 instead of λ, and the likelihood peak would still be in the same place.

d. In our experiment on Carbon-15 in Section 40.4.1, we observed k = 50 decay times and obtained a sample mean of x = 3.36. Write down your estimate of the half-life of Carbon-15, and suggest a confidence interval. (Hint: your half-life estimator is an MLE, so you can use the usual calculation of estimate ±2se for your confidence interval.)