Your company markets a disposable butane lighter called "surelight." In your product advertising, you use the slogan "lights on the first try-everytime!" As a quality check, you intend to examine a random sample of 10,000 lighters from the assembly line and observe for each lighter the number of trials required for the lighter to light. Your assistant obtains the random sample outcome and reports to you that a total of 10,118 trials were required to get all of the lighters to light. She did not record the 10,000 individual outcomes of how many trials were required for each lighter to light. You are interested in estimating both the expected number of trials needed for a lighter to light and the probability, \(p\), that the lighter lights on any given trial.

(a) Define an appropriate statistical model for the 10,000 outcomes of how many trials were required for each lighter to light.

(b) Define the MLE for the expected number of trials needed for a lighter to light. Is the estimator the MVUE? Is it consistent? Is it asymptotically normal? Is asymptotically efficient?

(c) Define the MLE for the probability that the lighter lights on any given trial. Is the estimator the MVUE? Is it consistent? Is it asymptotically normal? Is it asymptotically efficient? (Hint: Can you show that \(t(\mathbf{X})=(n-1) /\left(\left(\sum_{i=1}^{n} X_{i}ight)-1ight)\) has an expectation equal to \(p\) ?)

(d) Provide MLE estimates and MVUE estimates of both the expected number of trials needed for the first light and the probability that the lighter lights on any given trial.