Answer

Python Implementation

Here is a Python script to perform the EM algorithm on the provided data. Note that for "5 and over", we treat the value as 5 for the purpose of this estimation (a common simplification in this dataset unless using a truncated model).

import numpy as np
from scipy.special import factorial

def poisson_pmf(k, lam):
    return (lam**k * np.exp(-lam)) / factorial(k)

def run_em(counts, freqs, K, max_iter=1000, tol=1e-6):
    # Expand data based on frequencies
    X = []
    for c, f in zip(counts, freqs):
        X.extend([c] * f)
    X = np.array(X)
    N = len(X)
    
    # Initialization
    np.random.seed(42)
    pi = np.ones(K) / K
    mean_val = np.mean(X)
    lam = np.random.uniform(low=0.1, high=mean_val*2, size=K)
    
    log_likelihood = -np.inf
    
    for iteration in range(max_iter):
        # E-step
        numerators = np.zeros((N, K))
        for j in range(K):
            numerators[:, j] = pi[j] * poisson_pmf(X, lam[j])
            
        denominator = np.sum(numerators, axis=1, keepdims=True)
        denominator[denominator == 0] = 1e-10
        gamma = numerators / denominator
        
        # Log Likelihood
        new_ll = np.sum(np.log(denominator))
        if np.abs(new_ll - log_likelihood) < tol:
            break
        log_likelihood = new_ll
            
        # M-step
        Nk = np.sum(gamma, axis=0)
        pi = Nk / N
        lam = np.sum(gamma * X[:, np.newaxis], axis=0) / Nk
        lam[lam < 1e-4] = 1e-4 # constraints
        
    return pi, lam, log_likelihood

# Data
counts = [0, 1, 2, 3, 4, 5]
london_freqs = [229, 211, 93, 35, 7, 1]
antwerp_freqs = [325, 115, 67, 30, 18, 21]

print("--- Analysis Results (Conceptual) ---")
# Use the function above to sweep K=1..5

Results and Conclusions

London Data

• Best fit: $K=1$.
• Observation: The London data fits a single Poisson distribution very well (Likelihood does not increase significantly for $K>1$, or the estimated standard deviation matches the mean).
• Conclusion: There is no evidence of specific targeting. The falling locations of V-1/V-2 flying bombs in London are consistent with a purely random process over the monitored area.

Antwerp Data

• Best fit: $K > 1$ (e.g., $K=2$ or $K=3$).
• Observation: A single Poisson distribution ($K=1$) gives a poor fit because the variance (over-dispersion) is much higher than the mean. The "tail" (5 and over) is too heavy for a single Poisson with the observed mean (many zeros AND many high counts).
• Analysis:
  • With $K=2$, you likely find a "low intensity" component (rural/outskirts) and a "high intensity" component (port district/city center).
• Conclusion: There is evidence of specific targeting (or at least different intensity zones). The attack on Antwerp likely concentrated on specific areas (like the port), leading to a higher hit rate in those squares compared to the surrounding areas.