To study this, one day his trusted assistant records the time between successive customer

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Suppose Joe owns a pizza shop. We know it’s the favorite shop of at least one STA238 student, so this question is of serious practical importance. Joe is concerned with the number of customers he serves during the lunch hour. To study this, one day his trusted assistant records the time between successive customers. Let Xi be the number of minutes until the ith customer enters Joe’s shop, measured from when the previous customer entered.

We model X1, . . . , Xn as independent and identically distributed random variables from the Exponential (λ0) distribution, which is sometimes an appropriate model for waiting times. (Note that we are using here an alternative parametrization of the Exponential distribution, which might be different than what you’ve seen before; throughout this assignment work with the Exponential density given below.) The parameter λ represents the average number of minutes Joe has to wait until the next customer enters his shop, measured from the time the previous customer entered. λ = 2 would be 2 minutes per customer, λ = 0.1 would be 10 customers per minute, and so on. Joe wants to estimate the true value of λ, λ0. The exponential density is fλ(xi) = λ−1 exp(−xi/λ).

 

1.Show that the Maximum Likelihood Estimator (MLE) for λ in this model is λ = X

2The waiting times data are posted on the assignment page on Quercus in the file assignment3-waiting.csv. Consider the following code. Describe in words, in full detail, the bootstrap algorithm  that  is  im- plemented  in  the  code.  Indicate the distribution that  is  being  estimated  by  the bootstrap distribution.

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