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The Poisson Distribution is a statistical tool used to predict the likelihood of events occurring within a certain time or space. It is characterized by its mean and variance, both equal to the rate parameter λ, making it a valuable model for analyzing discrete events in fields like biology, management, and environmental science. Its probability mass function (PMF) and the relationship between mean and variance provide insights into event frequencies and their dispersion, aiding in practical applications such as inventory control, network traffic, and natural disaster predictions.

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## Definition and Properties of the Poisson Distribution

### Probability Mass Function (PMF)

The PMF is a function that gives the probability of a discrete random variable being equal to a specific value

### Mean and Variance

Mean

The mean, denoted by \( \mu \), is the average number of events expected to occur in a given interval

Variance

The variance, denoted by \( \sigma^2 \), measures the spread of event counts around the mean and is equal to the mean

### Properties

The Poisson Distribution is characterized by independent events, a constant mean rate, and equal mean and variance

## Applications of the Poisson Distribution

### Practical Uses

The Poisson Distribution is applied in various fields, such as biology, astronomy, and management, to model discrete events with a constant mean rate

### Statistical Significance

The Poisson Distribution's ability to quantify the probability of events within a defined interval makes it a crucial tool in theoretical research and practical problem-solving

### Advanced Applications

The Poisson Distribution is used in areas that require accurate modeling of discrete events, such as health sciences and traffic engineering

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