The problem involves finding Poisson probabilities related to homicide data over several city-years. From the image, the key parameters are as follows:

• There are 280 city-years in total.
• There were 73 homicides in total.

Step 1: Find the mean number of homicides per city-year.

The mean (λ) is calculated as the total number of homicides divided by the number of city-years: $\lambda = \frac{73}{280} = 0.2607$

Step 2: Poisson probability formula

The Poisson probability formula is: $P(k; \lambda) = \frac{e^{-\lambda} \lambda^k}{k!}$ Where:

• $k$ is the number of homicides.
• $\lambda = 0.2607$
• $e$ is the Euler's number (approximately 2.71828).

Step 3: Calculation for each value of $k$

We already have probabilities for $k = 0$, $k = 1$, and $k = 2$, so we can calculate for $k = 3$.

a. $P(0)$

$P(0) = \frac{e^{-0.2607} \cdot 0.2607^0}{0!} = e^{-0.2607} \approx 0.7705 \quad \text{(given)}$

b. $P(1)$

$P(1) = \frac{e^{-0.2607} \cdot 0.2607^1}{1!} \approx 0.2009 \quad \text{(given)}$

c. $P(2)$

$P(2) = \frac{e^{-0.2607} \cdot 0.2607^2}{2!} \approx 0.0262 \quad \text{(given)}$

d. $P(3)$

$P(3) = \frac{e^{-0.2607} \cdot 0.2607^3}{3!}$ Let's calculate this: $P(3) = \frac{e^{-0.2607} \cdot (0.2607)^3}{6} = \frac{e^{-0.2607} \cdot 0.01773}{6} \approx \frac{0.7705 \cdot 0.01773}{6} \approx \frac{0.01367}{6} \approx 0.0023$

Thus, $P(3) \approx 0.0023$.

Summary of probabilities:

• $P(0) = 0.7705$
• $P(1) = 0.2009$
• $P(2) = 0.0262$
• $P(3) = 0.0023$

Would you like more details or have any further questions?

Here are five related questions to consider:

1. How do Poisson distributions apply to real-world crime data analysis?
2. What assumptions are made when using a Poisson distribution for modeling homicides?
3. How does changing the mean $\lambda$ affect the shape of the Poisson distribution?
4. What would the probabilities look like for values of $k \geq 4$?
5. How could the accuracy of these probability models be verified with further data?

Tip: When working with small means like $\lambda = 0.2607$, the probabilities drop off quickly for higher values of $k$, which is why $P(3)$ and higher are very small.