STA360 at Duke University
Physicists studying a radioactive substance measure the times at which the substance emits a particle. They will record \(n+1\) emissions and set \(Y_1\) to be the time elapsed between the first and second emission, \(Y_2\) to be the time elapsed between the second and third emission and so on. They will model the data as \(Y_1, \ldots Y_n | \theta \sim \text{exponential}(\theta)\). The pdf of the exponential(\(\theta\)) distribution is
\[ p(y |\theta) = \theta e^{-\theta y} \ \text{ for } \ y>0, \ \theta>0. \]
For this distribution, \(E[Y|\theta] = \frac{1}{\theta}\).
(a). Write out the corresponding joint density \(p(y_1, \ldots, y_n | \theta)\) and simplify as much as possible. Justify each step of your calculation.
(b). For fixed values of \(y_1, \ldots y_n\), find the value of \(\theta\) that maximizes \(p(y_1, \ldots, y_n | \theta)\) as a function of \(\theta\), and call this maximizing value \(\hat{\theta}\). Hint: \(\hat{\theta}\) can also be found by maximizing \(\log p(y_1, \ldots, y_n | \theta)\), which is easier to work with.
(c). Suppose your information about \(\theta\) can be described with a gamma(\(a, b\)) prior distribution for some values of \(a\) and \(b\). Write out the formula for \(p(\theta | y_1, \ldots y_n)\), up to a proportionality in \(\theta\), and simplify as much as possible. From this, identify explicitly the posterior distribution of \(\theta\) (i.e., write “the posterior is a blank distribution with parameter(s) blank)”.
(d). Obtain the formula for \(E[\theta, y_1, \ldots y_n]\) as a function of \(a, b n\) and \(y_1, \ldots y_n\), and try to write this as a function of the estimator \(\hat{\theta}\) you found in part (b). What does \(E[\theta | y_1,\ldots,y_n]\) get close to as \(n\) increases?