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 𝜇. 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 𝜇 based on the data above.

## Answer<br /><br />The 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 𝜇 (mean number of claims per boat per year) can be calculated as the total number of claims divided by the total number of boats.<br /><br />Let's calculate the MLE for 𝜇 based on the given data.<br /><br />### Step 1: Calculate Total Number of Claims and Total Number of Boats<br /><br />| Year | Number of A Type boats insured | Number of claims |<br />|------|--------------------------------|-----------------|<br />| 1 | 80 | 12 |<br />| 2 | 98 | 21 |<br />| 3 | 145 | 26 |<br />| 4 | 158 | 38 |<br />| 5 | 166 | 52 |<br />| Total| 647 | 149 |<br /><br />### Step 2: Calculate MLE for 𝜇<br /><br />The MLE for 𝜇 is calculated as:<br /><br />```<br />𝜇 = Total Number of Claims / Total Number of Boats<br />```<br /><br />Substituting the values from the table:<br /><br />```<br />𝜇 = 149 / 647 ≈ 0.23<br />```<br /><br />So, the maximum likelihood estimate of 𝜇 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.