16.4 Mean and Variance of 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. A key characteristic of the Poisson distribution is that its mean and variance are equal.

Key Characteristics

In a Poisson distribution, the parameter $\lambda$ (lambda) represents the average rate of events occurring within the specified interval. This parameter $\lambda$ directly dictates both the mean (expected value) and the variance of the distribution.

Mean (Expected Value)

The expected value, or mean, of a Poisson distribution is the average number of events expected to occur in the given interval.

Using statistical notation: $$ E(X) = \mu = \lambda $$

Where:

• $E(X)$ is the expected value of the random variable $X$.
• $\mu$ (mu) is the symbol for the mean.
• $\lambda$ (lambda) is the rate parameter of the Poisson distribution.

Variance

The variance of a Poisson distribution measures the spread or dispersion of the distribution around its mean.

Using statistical notation: $$ Var(X) = \sigma^2 = \lambda $$

Where:

• $Var(X)$ is the variance of the random variable $X$.
• $\sigma^2$ (sigma squared) is the symbol for variance.
• $\lambda$ (lambda) is the rate parameter of the Poisson distribution.

The Equality of Mean and Variance

The defining property of the Poisson distribution is the equality of its mean and variance:

$$ E(X) = Var(X) = \lambda $$

This property makes the Poisson distribution particularly useful for modeling count data where the average number of occurrences is directly related to the variability of those occurrences. For instance, if you observe an average of 5 events per hour, you would also expect the variability around that average to be around 5.

Interpretation of $\lambda$

The parameter $\lambda$ is crucial and can be interpreted as:

• The average number of events occurring in a specific interval (time, space, volume, etc.).
• The rate parameter that governs the probability of observing a certain number of events.

Example

Consider a call center that receives an average of 10 calls per hour. If the number of calls follows a Poisson distribution, then:

• The mean number of calls per hour is $\lambda = 10$.
• The variance of the number of calls per hour is also $\lambda = 10$.

This implies that while the center can expect to receive around 10 calls per hour, the actual number of calls might deviate from this average, and the extent of this deviation is also quantified by 10.

Implications and Applications

The equal mean and variance property of the Poisson distribution has significant implications:

• Modeling Count Data: It's ideal for situations where events are rare but occur at a constant average rate, such as the number of defects per meter of fabric, the number of customers arriving at a store per minute, or the number of emails received per hour.
• Statistical Inference: This property simplifies statistical inference. If you have a sample of Poisson data, the sample mean and sample variance will tend to be close to each other.
• Model Fit: If the variance of your observed count data is significantly larger than its mean, a standard Poisson distribution might not be the best fit, and you might consider overdispersed Poisson models like the Negative Binomial distribution. Conversely, if the variance is much smaller than the mean, it suggests a different underlying process.

Interview Questions

• What is the mean of a Poisson distribution? The mean (expected value) of a Poisson distribution is equal to its rate parameter, $\lambda$.
• Why is the mean equal to the variance in a Poisson distribution? This equality arises from the underlying assumptions of the Poisson process, where events occur independently at a constant average rate. The mathematical derivation of the Poisson probability mass function leads to this characteristic relationship between the mean and variance.
• How is $\lambda$ interpreted in a Poisson distribution? $\lambda$ represents the average number of events that occur in a fixed interval of time or space. It is also the rate parameter of the distribution.
• What are the implications of $E(X) = Var(X) = \lambda$ in real-world applications? This implies that the spread or variability of the counts is directly proportional to the average rate of occurrence. Higher average rates lead to both higher expected counts and greater variability in those counts.
• Can the Poisson distribution be used if variance differs from the mean? Why or why not? No, if the variance significantly differs from the mean, a standard Poisson distribution is likely not appropriate. This situation is known as overdispersion (variance > mean) or underdispersion (variance < mean). For overdispersion, models like the Negative Binomial distribution are often used.
• In which scenarios is the Poisson distribution an appropriate model? It's appropriate for modeling the number of events occurring in a fixed interval when:
  • Events occur independently.
  • The average rate of occurrence ($\lambda$) is constant.
  • The probability of an event is proportional to the length of the interval.
  • No more than one event can occur in an infinitesimally small interval.
• How do you compute the mean and variance from sample Poisson data? The sample mean ($\bar{x}$) is calculated by summing all observed values and dividing by the number of observations. The sample variance ($s^2$) is calculated using the standard formula for sample variance. For a Poisson distribution, both $\bar{x}$ and $s^2$ are estimators of $\lambda$.
• Why is the Poisson distribution suitable for modeling random events over time? It's suitable because it captures the essence of random, independent occurrences happening at a steady average rate over a period, like customer arrivals, radioactive decays, or equipment failures.
• How does the value of $\lambda$ affect the shape of a Poisson distribution?
  • For small $\lambda$ (e.g., $\lambda < 5$), the distribution is highly skewed to the right (positively skewed).
  • As $\lambda$ increases, the distribution becomes more symmetric and bell-shaped, resembling a normal distribution.
• What distinguishes the Poisson distribution from other discrete distributions based on mean and variance? The key differentiator is the strict equality $E(X) = Var(X) = \lambda$. For example, the Binomial distribution has mean $np$ and variance $np(1-p)$, where the variance is always less than or equal to the mean. The Geometric distribution has mean $1/p$ and variance $(1-p)/p^2$, where the variance is greater than the mean.

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