Question

The mean and variance of a Poisson distribution are

Short answer

Mean and variance of a Poisson distribution are both \( \lambda \).

Step-by-step solution

Understanding Poisson Distribution

Before proceeding, know that a Poisson distribution is used to model the number of events occurring within a fixed period of time or space if these events happen with a known constant mean rate and independently of the time since the last event.

Identifying the Mean of the Poisson Distribution

The parameter of a Poisson distribution, typically denoted by \( \lambda \), represents the average number of events in the given interval. Consequently, the mean of a Poisson distribution is \( \lambda \).

Identifying the Variance of the Poisson Distribution

In a Poisson distribution, the variance is equal to its mean. Thus, the variance is also \( \lambda \). This reflects the property that in a Poisson distribution, variance and mean are numerically equivalent.

Key concepts

Mean in Poisson Distribution

In a Poisson distribution, the mean is a crucial concept. It's denoted by the Greek letter \( \lambda \), which represents the average number of occurrences of a given event within a specific time frame or spatial area. For example, if you're studying the number of emails received per hour, \( \lambda \) would be the average number of emails you receive in one hour.
This parameter is essential because it indicates the rate at which events are expected to happen. It serves as the foundation for predicting and analyzing patterns within the data. Knowing the mean also helps in determining expectations for future events.
Notably, in a practical sense, knowing the mean allows us to anticipate and prepare for scenarios where the event count exceeds or falls below this average, thus making this concept valuable for forecasting in various fields.

Variance in Poisson Distribution

A fascinating characteristic of the Poisson distribution is that its variance is equal to its mean. This means that the spread or dispersion of the data within the Poisson distribution is represented by the same \( \lambda \) parameter as the mean.
This aspect reflects the predictability and precision of the Poisson model, especially because variance usually measures how much values differ from the mean. Here, since variance equals the mean, understanding and predicting patterns becomes more feasible.
Practically, having the variance equal to the mean implies that as \( \lambda \) increases, the distribution becomes more spread out. This is due to the increased possibility of different outcomes as the average number of events per interval grows.
This property of variance being equal to the mean is unique to the Poisson distribution and is not commonly found in other types of probability distributions.

Properties of Poisson Distribution

The Poisson distribution has several unique properties that make it useful in statistical modeling. These properties assist in understanding how and when to appropriately apply this distribution.

• Independence: Events are independent of each other. This means that the occurrence of one event does not affect the probability of another event occurring.
• Consistency: The distribution remains consistent, with both mean and variance equal to \( \lambda \). This property offers simplicity in calculating dispersion.
• Non-negative integers: The possible outcomes are whole, non-negative numbers, like 0, 1, 2, 3, making it ideal for counting occurrences.
• Scalability: As the observation period changes, the rate \( \lambda \) can be adjusted, providing flexibility for different time frames.
• Memoryless: The distribution does not require information from previous intervals, allowing fresh analysis each time events are counted.

These properties make the Poisson distribution suitable for modeling events such as customer arrivals, phone call frequencies, or emails received. Understanding these properties can help ensure the correct application and interpretation of this statistical model.