To assess the likelihood of the two hypotheses regarding the occurrence of the five criminal events over a 23-month period versus the last 6 months, we can use a Poisson process framework. The Poisson process is a common statistical model for events that occur independently over time.
Defining the parameters
We assume that the events (assaults, break-ins, thefts) are independent and follow a Poisson distribution. The key parameter for the Poisson distribution is the rate \( \lambda \) of events per unit time (in this case, per month).
Given that there have been 5 events in the last 6 months, we can calculate the average rate of events per month based on this time frame.
Step 1: Calculate the rate \( \lambda \)
If 5 events occurred in the last 6 months, the average event rate (\( \lambda \)) can be calculated as:
\[ \lambda = \frac{5 \text{ events}}{6 \text{ months}} \approx 0.833 \text{ events/month} \]
Step 2: Calculate the expected number of events over 23 months using this rate
For a 23-month period, the expected number of events when the rate is 0.833 events/month is:
\[ E_{23} = \lambda \times 23 \approx 0.833 \times 23 \approx 19.166 \text{ events} \]
Poisson probability calculations
Step 3: Calculate the probability of observing 5 or more events in 6 months versus 23 months
- For the last 6 months: We need to calculate the probability of observing 5 events under the Poisson distribution:
\[ P(X = k) = \frac{e^{-\lambda t} (\lambda t)^k}{k!} \]
Where \( t \) is time (6 months here) and \( k \) is the count of events (5 events here). For \( k = 5 \):
\[ P(X = 5) = \frac{e^{-5} \cdot 5^5}{5!} \approx \frac{e^{-5} \cdot 3125}{120} \approx 0.1755 \]
- For 23 months: Similarly, we can compute the probability of observing 5 events in a 23-month period:
For \( t = 23 \), using \( \lambda t \approx 19.166 \):
We calculate the probability of observing 0 to 5 events (since we want to show that it’s very unlikely).
Using the Poisson cumulative distribution function (CDF), we need to calculate:
\[ P(X \leq 5) = \sum_{k=0}^{5} \frac{e^{-19.166} \cdot 19.166^k}{k!} \]
Calculating for each \( k \) up to 5 manually or using statistical software:
- For \( k=0 \): \( e^{-19.166} \)
- For \( k=1 \): \( \frac{19.166}{1!} \cdot e^{-19.166} \)
- For \( k=2 \): \( \frac{19.166^2}{2!} \cdot e^{-19.166} \)
- Continue up to \( k=5 \).
This will yield a very small cumulative probability, indicating that seeing 5 specific events in 23 months is very unlikely.
Conclusion
- The likelihood of experiencing 5 events under a consistent average rate over 23 months is astronomically low.
- Therefore, the occurrence of these events in the last 6 months suggests they are likely due to non-random factors rather than random chance. In terms of exact figures, you would want to compute the CDF for clarity but expect it to be around several orders of magnitude lower compared to events clustered in a small timeframe.
In summary, it is significantly more likely that something caused these events than they are a result of random occurrences.