Poisson random variable

Definition

A Poisson random variable is a type of discrete random variable that represents the number of events occurring within a fixed interval of time or space, given that these events occur with a known constant mean rate and independently of the time since the last event. It’s widely used in scenarios where events happen randomly and independently, like phone calls received at a call center or the number of decay events from a radioactive source over time.

5 Must Know Facts For Your Next Test

1. The probability mass function for a Poisson random variable is given by the formula: $$P(X=k) = \frac{e^{-\lambda} \lambda^k}{k!}$$, where $$k$$ is the number of events, $$e$$ is Euler's number, and $$\lambda$$ is the average rate.
2. The mean and variance of a Poisson random variable are both equal to $$\lambda$$, making it unique among discrete distributions.
3. Poisson random variables can model real-life scenarios like the number of emails received in an hour or customer arrivals at a store in a day.
4. As $$\lambda$$ increases, the Poisson distribution begins to resemble a normal distribution due to the Central Limit Theorem.
5. In practice, a Poisson distribution is often used when considering rare events that occur independently over time or space.

Review Questions

• How does the Poisson random variable differ from other discrete random variables in terms of its application and properties?
  • The Poisson random variable is specifically used to model counts of independent events occurring within a fixed interval. Unlike other discrete random variables that might have varying probabilities, the Poisson variable has a consistent average rate (lambda) which governs its behavior. Additionally, while some discrete variables can take on any non-negative integer value with differing probabilities, the Poisson's properties make it suitable for scenarios where events occur randomly over time or space.
• In what situations would you prefer using a Poisson distribution over a binomial distribution, and why?
  • You would prefer using a Poisson distribution over a binomial distribution when dealing with rare events occurring over a large number of trials or when the number of trials is unknown. The Poisson model assumes an infinite number of possible trials while maintaining a constant average event rate. In contrast, the binomial distribution is more appropriate when you have a fixed number of trials and a defined probability for each success. Thus, for scenarios like phone call arrivals where each call is independent and there isn't a fixed limit on calls, the Poisson distribution fits better.
• Critically evaluate how changing the value of lambda (λ) affects the shape and characteristics of the Poisson distribution.
  • Changing the value of lambda (λ) has significant effects on the shape and characteristics of the Poisson distribution. As λ increases, both the mean and variance increase, leading to a shift in the distribution towards higher values. The shape transitions from being skewed to more symmetric, resembling a normal distribution as λ becomes larger. Conversely, if λ is small, the distribution remains highly skewed towards zero. This means that with low λ values, outcomes with fewer events are more likely than outcomes with many events, illustrating how λ directly influences event frequency modeling in various applications.

Related terms

Lambda (λ): The average rate at which events occur in a Poisson distribution, representing the expected number of occurrences in the interval.

Exponential distribution:

A continuous probability distribution that describes the time between events in a Poisson process, where events occur continuously and independently.

Binomial distribution:

A discrete probability distribution that models the number of successes in a fixed number of trials, often compared to the Poisson distribution when the number of trials is large and the probability of success is small.