Problem 2.1 (c)

Pre-required Knowledge

• Maximum Likelihood Estimate formula: From part (a), we know $\hat{\lambda} = \frac{1}{N} \sum_{i=1}^{N} k_i$. This corresponds to the total number of hits divided by the total number of cells.

Step-by-Step Answer

1. Identify the Total Number of Cells ($N$): Sum the number of cells for each category: $N = 229 + 211 + 93 + 35 + 7 + 1 = 576$

2. Calculate the Total Number of Hits ($\sum k_i$): Multiply the number of hits ($k$) by the count of cells with that many hits. Note: For the category "5 and over", we assume the number of hits is exactly 5 based on the standard interpretation of this dataset in literature, as higher counts are extremely unlikely.

   $$\begin{aligned} \text{Sum} &= (0 \times 229) + (1 \times 211) + (2 \times 93) + (3 \times 35) + (4 \times 7) + (5 \times 1) \\ &= 0 + 211 + 186 + 105 + 28 + 5 \\ &= 535 \end{aligned}$$

3. Calculate $\hat{\lambda}$:

   $$\hat{\lambda} = \frac{\text{Total Hits}}{\text{Total Cells}} = \frac{535}{576} \approx 0.928819...$$

4. Result: $\hat{\lambda} \approx 0.929$