Understanding the Poisson Distribution: Calculating Accidents in a Month

In a given city, the number of monthly accidents on a particular street is known to follow the Poisson distribution with a mean value of 3 accidents per month. The Poisson distribution is a statistical tool that is often used to model the number of events (such as accidents, phone calls, or customer arrivals) occurring in a fixed interval of time or space. This article will guide you through calculating the probability of exactly 2 accidents happening in a given month using the Poisson distribution.

Introduction to the Poisson Distribution

The Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and independently of the time since the last event. The distribution is named after French mathematician Siméon Denis Poisson.

The only parameter required for the Poisson distribution is λ (lambda), which represents the average number of events in the interval. In this case, the average number of accidents per month (λ) is 3.

The Poisson Probability Formula

The formula for the Poisson probability mass function (PMF) is given by:

P(X k) (λ^k * e^(-λ)) / k!

Where:

X is the number of events (in our case, accidents).

λ is the average number of events (3 accidents per month).

k is the number of events we are interested in (2 accidents in this case).

e is the base of the natural logarithm, approximately equal to 2.71828.

! denotes the factorial function, for example, 2! 2 × 1 2 and 1! 1.

Calculating the Probability

To find the probability of exactly 2 accidents in a given month, we need to substitute λ 3 and k 2 into the Poisson probability formula:

P(X 2) (3^2 * e^(-3)) / 2!

Step 1: Calculate 3^2

3^2 9

Step 2: Calculate e^(-3)

e^(-3) ≈ 0.04979

Step 3: Calculate the factorial 2!

2! 2 × 1 2

Step 4: Substitute these values into the formula:

P(X 2) (9 * 0.04979) / 2

P(X 2) ≈ (0.44811) / 2

P(X 2) ≈ 0.224055

Therefore, the probability of exactly 2 accidents in a given month is approximately 0.224 (or 22.4%).

Real-World Applications of the Poisson Distribution

The Poisson distribution is widely used in various fields, including:

Quality control in manufacturing to predict the number of defective products.

Telecommunications to forecast the number of calls received by a call center.

Healthcare to estimate the number of patients visiting a clinic.

Finance to model the number of insurance claims.

Conclusion

The Poisson distribution is a powerful tool for analyzing and predicting events that occur at a constant rate. By understanding the properties of the distribution and its parameters, we can effectively calculate probabilities associated with specific events. In the case of monthly accidents on a street, we have determined the probability of exactly 2 accidents in one month, providing valuable insights for urban planners and safety officers.

For more information on the Poisson distribution and its applications, or if you need to calculate probabilities for other values of λ and k, feel free to explore further resources or seek assistance from a statistician.