The Poisson distribution has been used by traffic engineers...

The Poisson distribution has been used by traffic engineers as a model for light traffic, based on the rationale that if the rate is approximately constant and the traffic is light (so the individual cars move independently of each other), the distribution of counts of cars in a given time interval or space area should be nearly Poisson (Gerlough and Schuhl 1955 ). The following table shows the number of right turns during 300 3-min intervals at a specific intersection. Fit a Poisson distribution. Comment on the fit by comparing observed and expected counts. It is useful to know that the 300 intervals were distributed over various hours of the day and various days of the week. $$\begin{array}{cc} \hline n & \text { Frequency } \\ \hline 0 & 14 \\ 1 & 30 \\ 2 & 36 \\ 3 & 68 \\ 4 & 43 \\ 5 & 43 \\ 6 & 30 \\ 7 & 14 \\ 8 & 10 \\ 9 & 6 \\ 10 & 4 \\ 11 & 1 \\ 12 & 1 \\ 13+ & 0 \\ \hline \end{array}$$

The Poisson distribution has been used by traffic engineers as a model for light traffic, based on the rationale that if the rate is approximately constant and the traffic is light (so the individual cars move independently of each other), the distribution of counts of cars in a given time interval or space area should be nearly Poisson (Gerlough and Schuhl 1955 ). The following table shows the number of right turns during 300 3-min intervals at a specific intersection. Fit a Poisson distribution. Comment on the fit by comparing observed and expected counts. It is useful to know that the 300 intervals were distributed over various hours of the day and various days of the week.
$$\begin{array}{cc}
\hline n & \text { Frequency } \\
\hline 0 & 14 \\
1 & 30 \\
2 & 36 \\
3 & 68 \\
4 & 43 \\
5 & 43 \\
6 & 30 \\
7 & 14 \\
8 & 10 \\
9 & 6 \\
10 & 4 \\
11 & 1 \\
12 & 1 \\
13+ & 0 \\
\hline
\end{array}$$

Key Concepts

Poisson Distribution

A Poisson distribution is a discrete probability distribution used to model the number of times an event occurs in a fixed interval of time or space, under the assumption that these events occur with a constant mean rate and independently. This makes it suitable for modeling rare events that happen at a predictable average rate.

Parameter Estimation (Lambda)

In a Poisson distribution, the parameter lambda (?) represents both the mean and the variance of the distribution. It is typically estimated by computing the average number of observed events over the intervals of interest. This estimate is crucial for generating the expected counts used in model comparisons.

Goodness-of-Fit Testing

Goodness-of-fit testing involves comparing the observed frequencies of events with the expected frequencies under the assumed Poisson model, often using a chi-squared test. This comparison helps in assessing whether the Poisson model is an appropriate fit for the data, indicating how well the model captures the underlying event distribution.

Modeling Assumptions

The application of the Poisson distribution relies on key assumptions: the events must occur independently, the event rate must be constant over the considered intervals, and the probability of multiple events in a very short interval is negligible. Violations of these assumptions can lead to discrepancies between observed data and the model predictions.

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