Question

The mean and variance of Poisson distribution are equal.

Short answer

In a Poisson distribution, mean equals variance, both equal to \(\lambda\).

Step-by-step solution

Understanding 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. The key parameter in Poisson distribution is \( \lambda \), which represents both the mean and the variance of the distribution.

Mean of Poisson Distribution

For a Poisson distribution, the mean (expected value) is given by \( E(X) = \lambda \). This means that the average number of events in the given interval is \( \lambda \).

Variance of Poisson Distribution

Similarly, the variance in a Poisson distribution is determined as \( \text{Var}(X) = \lambda \). This means the variability or dispersion of the number of events around the mean is also \( \lambda \).

Conclusion: Mean Equals Variance

The key property of the Poisson distribution is that its mean and variance are equal. Therefore, \( E(X) = \text{Var}(X) = \lambda \). This property simplifies many calculations involving Poisson distributions.

Key concepts

Discrete Probability Distribution

The Poisson distribution is a type of discrete probability distribution. This means it deals with events that can be counted and occur individually, rather than continuously. In simple terms, think of it as the distribution that counts the number of times an event happens within a specified timeframe or space. For instance, it might measure the number of cars passing through a checkpoint per hour or how many emails you receive in a day. In the realm of probability, distributions like the Poisson provide a mathematical way to predict or explain such events. Discrete indicates that we're dealing with distinct, separate events. The opposite would be a continuous probability distribution, which applies to data that can be any value within a range, like height or weight. Understanding this distinction is vital because it guides how we analyze data. With the Poisson distribution, we focus on probabilities of a specific number of occurrences, making it a handy tool in fields like telecommunications, epidemiology, and queueing theory.

Mean and Variance

In the Poisson distribution, both the mean and variance hold a unique relationship – they are equal, both represented by the parameter \( \lambda \). Let's break these terms down:

• Mean: The mean, or expected value, \( E(X) \), signifies the average outcome you would anticipate over a long period. In Poisson terms, if you are observing the number of occurrences, \( \lambda \) represents this expected average. Hence, it gives us a sense of centrality of the distribution.
• Variance: Variance, denoted as \( \text{Var}(X) \), measures the spread or dispersion of the outcomes around the mean. In a Poisson distribution, this spread is also governed by \( \lambda \). Hence, knowing \( \lambda \) tells us both the average number of occurrences and how much variation there is in these numbers.

This equality of mean and variance is a critical property, simplifying calculations and making predictions about the data's behavior more straightforward. For instance, when modeling the number of rare events like machine failures or email arrivals, having identical mean and variance provides significant insight into the expected spread of occurrences.

Parameter Lambda

Lambda, \( \lambda \), is the cornerstone of the Poisson distribution. This parameter not only defines the distribution but also equates its mean and variance. Here’s what you need to know:

• Definition: \( \lambda \) represents the average rate at which the events occur within the defined interval. It's the key input required to calculate probabilities in a Poisson distribution.
• Role in the Distribution: In practical applications, knowing \( \lambda \) enables you to predict the probability of a certain number of events happening. For example, if \( \lambda \) is 4 for the number of buses arriving in an hour, it tells us that on average, 4 buses are expected. The probabilities of seeing 0, 1, 2, 3,..., buses can then be calculated.
• Impact on Shape: As \( \lambda \) increases, the distribution tends to spread out and become more symmetric. With a smaller \( \lambda \), it is skewed towards zero, implying fewer frequent occurrences.

Understanding \( \lambda \) is crucial in using the Poisson distribution effectively, as it underpins the calculation of probabilities and reflects the characteristics of the dataset being modeled.