Characteristics of Poisson distribution

The Poisson distribution is a probability distribution that describes the number of events occurring in a fixed interval of time or space, given the average rate of occurrence and assuming independence between events.

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Here are some characteristics of the Poisson distribution:

1. Parameter: The Poisson distribution has one parameter, denoted by λ (lambda), which represents the average rate of occurrence of the events within the specified interval.
2. Discrete Distribution: The Poisson distribution is a discrete probability distribution, meaning it describes the probability of observing a certain number of events (usually non-negative integers) within a given interval.
3. Independence: The occurrences of events in a Poisson process are assumed to be independent of each other. This means that the probability of an event occurring does not affect the probability of other events occurring within the same interval.
4. Memorylessness: The Poisson distribution exhibits the property of memorylessness, which means that the probability of an event occurring in the future does not depend on how much time has already passed since the last event. In other words, past events do not influence the probability of future events.
5. Mean and Variance: The mean (μ) and variance (σ^2) of a Poisson distribution are both equal to λ, the parameter representing the average rate of occurrence. Mathematically, μ = λ and σ^2 = λ.
6. Probability Mass Function (PMF): The probability mass function of the Poisson distribution gives the probability of observing a specific number of events (k) within the interval. It is given by the formula: P(X = k) = (e^(-λ) * λ^k) / k!, where:

• e is the base of the natural logarithm (approximately 2.71828),
• λ is the average rate of occurrence,
• k is the number of events observed, and
• k! represents the factorial of k.

1. Applicability: The Poisson distribution is commonly used to model rare events occurring in a fixed interval of time or space, such as the number of phone calls received by a call center in an hour, the number of accidents at a particular intersection in a day, or the number of mutations in a DNA sequence.

Overall, the Poisson distribution is a fundamental tool in probability theory and statistics, particularly for modeling count data in various real-world applications.