Question
A general insurance company offers policies for three different kinds of rowing boat: A Type, B Type and C Type. For A Type boats, the number of claims per boat per year follows a Poisson distribution with parameter π. The following data is available for the past five years: Year 1 2 3 4 5 Number of A Type boats insured 80 98 145 158 166 Number of claims 12 21 26 38 52 (i) Derive the maximum likelihood estimate of π based on the data above.
Answer
4.7
(1 Votes)
Nesta
Master Β· Tutor for 5 years
Answer
## AnswerThe maximum likelihood estimate (MLE) of a parameter is the value that maximizes the likelihood function. In the case of a Poisson distribution, the MLE of the parameter π (mean number of claims per boat per year) can be calculated as the total number of claims divided by the total number of boats.Let's calculate the MLE for π based on the given data.### Step 1: Calculate Total Number of Claims and Total Number of Boats| Year | Number of A Type boats insured | Number of claims ||------|--------------------------------|-----------------|| 1 | 80 | 12 || 2 | 98 | 21 || 3 | 145 | 26 || 4 | 158 | 38 || 5 | 166 | 52 || Total| 647 | 149 |### Step 2: Calculate MLE for πThe MLE for π is calculated as:```π = Total Number of Claims / Total Number of Boats```Substituting the values from the table:```π = 149 / 647 β 0.23```So, the maximum likelihood estimate of π based on the data above is approximately 0.23. This means that, on average, there are about 0.23 claims per boat per year for A Type boats.