Problem 4 - A Modern Introduction to Probability and Statistics: Understanding Why and How

Question

During the Second World War, London was hit by numerous flying bombs. The following data are from an area in South London of 36 square kilometers. The area was divided into 576 squares with sides of length \(1 / 4\) kilometer. For each of the 576 squares the number of hits was recorded. In this way we obtain a dataset \(x_{1}, x_{2}, \ldots, x_{576}\), where \(x_{i}\) denotes the number of hits in the \(i\) th square. The data are summarized in the following table which lists the number of squares with no hits, 1 hit, 2 hits, etc. $$ \begin{array}{lcccccccc} \hline \hline \text { Number of hits } & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \text { Number of squares } & 229 & 211 & 93 & 35 & 7 & 0 & 0 & 1 \\ \hline \hline \end{array} $$ An interesting question is whether London was hit in a completely random manner. In that case a Poisson distribution should fit the data. a. If we model the dataset as the realization of a random sample from a Poisson distribution with parameter \(\mu\), then what would you choose as an estimate for \(\mu\) ? b. Check the fit with a Poisson distribution by comparing some of the observed relative frequencies of 0 's, 1 's, 2 's, etc., with the corresponding probabilities for the Poisson distribution with \(\mu\) estimated as in part a.

Step-by-Step Solution

Answer

Estimate \(\mu\) as 0.9323 and compare observed relative frequencies to Poisson probabilities.

Step 1: Calculate Total Hits

First, calculate the total number of hits across all 576 squares. Multiply the number of hits by the number of squares for each category and sum them up: \[(0 \times 229) + (1 \times 211) + (2 \times 93) + (3 \times 35) + (4 \times 7) + (5 \times 0) + (6 \times 0) + (7 \times 1) = 0 + 211 + 186 + 105 + 28 + 0 + 0 + 7 = 537\]

Step 2: Calculate Estimate for \(\mu\)

The parameter \(\mu\) of a Poisson distribution is estimated as the sample mean of the data. Divide the total number of hits by the total number of squares to find \(\mu\). \[\mu = \frac{537}{576} \approx 0.9323\]

Step 3: Calculate Poisson Probabilities

For each observed number of hits, calculate the corresponding Poisson probability using the estimated \(\mu = 0.9323\). The probability of observing \(k\) hits is given by \[ P(X = k) = \frac{e^{-\mu} \mu^k}{k!} \] Calculate these probabilities for 0, 1, 2, 3, and 4 hits.

Step 4: Calculate Relative Frequencies of Observed Hits

Calculate the relative frequency of each number of hits. Divide the number of squares for each category by the total number of squares (576).\[P(0) = \frac{229}{576}, \quad P(1) = \frac{211}{576}, \quad P(2) = \frac{93}{576}, \quad P(3) = \frac{35}{576}, \quad P(4) = \frac{7}{576}\]

Step 5: Compare Relative Frequencies with Poisson Probabilities

Compare the calculated Poisson probabilities with the relative frequencies determined in step 4. If the values are close, the Poisson distribution is a good fit. Otherwise, it indicates a poor fit.

Key Concepts

Random Sample

A random sample is a collection of observations drawn from a larger population, where every member of the population has an equal chance of being included in the sample. This randomness ensures that the sample fairly represents the overall group, avoiding biases and the introduction of systematic errors. In our context of the London bombing dataset, we can assume each square is part of a random sample from the total area hit by flying bombs. Understanding how to work with random samples is crucial because it affects the reliability of conclusions made from the data.

• Randomness ensures every unit or individual from the population deserves equal opportunity to be chosen.
• Maintaining randomness helps in making generalizations about the broader population from which the sample was drawn.

Without randomness, the data might reflect unintended patterns due to biases, affecting the results of any statistical analysis we perform, such as fitting the Poisson distribution to our dataset.

Parameter Estimation

Parameter estimation involves determining the parameters of a statistical model that best explains the data. In the Poisson distribution context, the parameter \( \mu \) represents the average rate of occurrence of events in a fixed interval of time or space.
To estimate \( \mu \) for the bombing data, we need to calculate the sample mean, which is the total number of hits divided by the total number of squares. This step gives us an estimate of the average number of hits per square. The formula used is:
\[ \mu = \frac{\text{Total Hits}}{\text{Total Squares}} \]
This estimate serves as a crucial step in applying the Poisson model to predict probabilities of different numbers of hits. Once \( \mu \) is estimated as 0.9323, it sharply informs the subsequent calculation of Poisson probabilities and plays a central role in evaluating the fit of the Poisson model to the observed data.

Probability Distribution

A probability distribution is a mathematical description that lists all possible outcomes of a random experiment and their associated probabilities. In this exercise, we focus on the Poisson distribution, a discrete probability distribution used to model the number of events happening in a fixed interval of time or space.
The Poisson distribution is defined by its parameter \( \mu \), which is both the mean and variance of the distribution. The probability of observing exactly \( k \) events is computed using the formula:
\[ P(X = k) = \frac{e^{-\mu} \mu^k}{k!} \]
This expression allows us to determine the likelihood of different numbers of hits, providing a theoretical framework to match with our actual data. When the calculated probabilities closely align with observed relative frequencies, it indicates a particularly good fit of the Poisson model to the dataset. This congruence helps us confirm that the distribution describes the random processes in the scenario accurately.

Relative Frequency

Relative frequency is a practical way to illustrate how often a particular outcome appears in a dataset relative to all possible outcomes. It's calculated by dividing the frequency of the specific event by the total number of observations.
In our example, we calculate the relative frequency for each number of hits over all the squares. For example, the relative frequency for 0 hits is calculated by dividing the number of squares with 0 hits by the total squares: \( P(0) = \frac{229}{576} \). This process is repeated for each hit count.

• Relative frequencies provide an intuitive understanding of the dataset's distribution.
• These frequencies can be directly compared to theoretical probabilities from the Poisson distribution.

By comparing these observed frequencies to the expected probabilities from our Poisson model, we can determine how well the data aligns with the model, thus providing insights into whether the Poisson distribution is a suitable representation of the random bombing events in London.