Confused About This Application Of The Poisson Limit Theorem -- How Can Λ = M N \lambda = \frac M N Λ = N M ​ ?

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Introduction

The Poisson Limit Theorem is a fundamental concept in probability theory that describes the limiting behavior of certain discrete distributions. It states that under certain conditions, the binomial distribution converges to the Poisson distribution as the number of trials increases. However, the application of this theorem can sometimes be confusing, especially when it comes to the calculation of the parameter $\lambda$. In this article, we will explore the reasoning behind the application of the Poisson Limit Theorem and clarify the calculation of $\lambda = \frac m n$.

Background on the Poisson Limit Theorem

The Poisson Limit Theorem is a result that describes the limiting behavior of the binomial distribution as the number of trials increases. The binomial distribution is a discrete distribution that models the number of successes in a fixed number of independent trials, each with a constant probability of success. The Poisson distribution, on the other hand, is a continuous distribution that models the number of events occurring in a fixed interval of time or space.

The Poisson Limit Theorem states that if we have a sequence of binomial distributions with parameters $n$ and $p$, where $n$ is the number of trials and $p$ is the probability of success, then as $n$ increases, the binomial distribution converges to the Poisson distribution with parameter $\lambda = np$.

The Application of the Poisson Limit Theorem

In the context of the problem you are working on, the Poisson Limit Theorem is invoked to show that $\lim _{m \rightarrow \infty} \operatorname{Pr}\left(X_m=r\right)=\frac{\mathrm{e}^{-m / n}(m / n)^r}{r!}$. However, you are confused about the reasoning used to calculate $\lambda = \frac m n$.

Calculating $\lambda$

To calculate $\lambda$, we need to understand the relationship between the binomial distribution and the Poisson distribution. In the binomial distribution, the parameter $n$ represents the number of trials, and the parameter $p$ represents the probability of success. In the Poisson distribution, the parameter $\lambda$ represents the average rate of events occurring in a fixed interval of time or space.

The key insight here is that the binomial distribution can be viewed as a sequence of independent Bernoulli trials, each with a probability of success $p$. The number of successes in $n$ trials is then a binomial random variable with parameters $n$ and $p$.

As $n$ increases, the binomial distribution converges to the Poisson distribution with parameter $\lambda = np$. This is because the binomial distribution can be approximated by a Poisson distribution with the same mean and variance.

Deriving the Limiting Distribution

To derive the limiting distribution, we can use the following steps:

1. Write down the probability mass function of the binomial distribution.
2. Take the limit as $n$ increases.
3. Simplify the resulting expression to obtain the Poisson distribution.

Using this approach, we can derive the limiting distribution as follows:

$$\lim _{n \rightarrow \infty} \operatorname{Pr}\left(X_n=r\right)=\lim _{n \rightarrow \infty} \binom{n}{r} p^r (1-p)^{n-r}$$

$$=\lim _{n \rightarrow \infty} \frac{n!}{r!(n-r)!} p^r (1-p)^{n-r}$$

$$=\lim _{n \rightarrow \infty} \frac{n(n-1)\cdots(n-r+1)}{r!} p^r (1-p)^{n-r}$$

$$=\lim _{n \rightarrow \infty} \frac{n^r}{r!} p^r (1-p)^{n-r}$$

$$=\frac{\mathrm{e}^{-p}(p)^r}{r!}$$

Conclusion

In conclusion, the application of the Poisson Limit Theorem involves calculating the parameter $\lambda = np$. This parameter represents the average rate of events occurring in a fixed interval of time or space. The limiting distribution is then obtained by taking the limit of the binomial distribution as $n$ increases.

The key insight here is that the binomial distribution can be viewed as a sequence of independent Bernoulli trials, each with a probability of success $p$. The number of successes in $n$ trials is then a binomial random variable with parameters $n$ and $p$.

As $n$ increases, the binomial distribution converges to the Poisson distribution with parameter $\lambda = np$. This is because the binomial distribution can be approximated by a Poisson distribution with the same mean and variance.

Frequently Asked Questions

Q: What is the Poisson Limit Theorem?

A: The Poisson Limit Theorem is a result that describes the limiting behavior of the binomial distribution as the number of trials increases.

Q: What is the parameter $\lambda$ in the Poisson distribution?

A: The parameter $\lambda$ represents the average rate of events occurring in a fixed interval of time or space.

Q: How is the parameter $\lambda$ calculated?

A: The parameter $\lambda$ is calculated as $\lambda = np$, where $n$ is the number of trials and $p$ is the probability of success.

Q: What is the relationship between the binomial distribution and the Poisson distribution?

A: The binomial distribution can be viewed as a sequence of independent Bernoulli trials, each with a probability of success $p$. The number of successes in $n$ trials is then a binomial random variable with parameters $n$ and $p$. As $n$ increases, the binomial distribution converges to the Poisson distribution with parameter $\lambda = np$.

References

• [1] Feller, W. (1968). An Introduction to Probability Theory and Its Applications. Wiley.
• [2] Grimmett, G. R., & Stirzaker, D. R. (2001). Probability and Random Processes. Oxford University Press.
• [3] Ross, S. M. (2010). Introduction to Probability Models. Academic Press.

Further Reading

• [1] Poisson Distribution on Wikipedia
• [2] Binomial Distribution on Wikipedia
• [3] Poisson Limit Theorem on MathWorld
  

Introduction

The Poisson Limit Theorem is a fundamental concept in probability theory that describes the limiting behavior of the binomial distribution as the number of trials increases. However, the application of this theorem can sometimes be confusing, especially when it comes to the calculation of the parameter $\lambda$. In this article, we will answer some frequently asked questions about the Poisson Limit Theorem and the binomial distribution.

Q&A

Q: What is the Poisson Limit Theorem?

A: The Poisson Limit Theorem is a result that describes the limiting behavior of the binomial distribution as the number of trials increases. It states that if we have a sequence of binomial distributions with parameters $n$ and $p$, where $n$ is the number of trials and $p$ is the probability of success, then as $n$ increases, the binomial distribution converges to the Poisson distribution with parameter $\lambda = np$.

Q: What is the parameter $\lambda$ in the Poisson distribution?

A: The parameter $\lambda$ represents the average rate of events occurring in a fixed interval of time or space. It is calculated as $\lambda = np$, where $n$ is the number of trials and $p$ is the probability of success.

Q: How is the parameter $\lambda$ calculated?

A: The parameter $\lambda$ is calculated as $\lambda = np$, where $n$ is the number of trials and $p$ is the probability of success.

Q: What is the relationship between the binomial distribution and the Poisson distribution?

A: The binomial distribution can be viewed as a sequence of independent Bernoulli trials, each with a probability of success $p$. The number of successes in $n$ trials is then a binomial random variable with parameters $n$ and $p$. As $n$ increases, the binomial distribution converges to the Poisson distribution with parameter $\lambda = np$.

Q: What is the Poisson distribution?

A: The Poisson distribution is a continuous distribution that models the number of events occurring in a fixed interval of time or space. It is characterized by a single parameter $\lambda$, which represents the average rate of events occurring in the interval.

Q: What is the binomial distribution?

A: The binomial distribution is a discrete distribution that models the number of successes in a fixed number of independent trials, each with a constant probability of success. It is characterized by two parameters $n$ and $p$, where $n$ is the number of trials and $p$ is the probability of success.

Q: How does the Poisson Limit Theorem apply to real-world problems?

A: The Poisson Limit Theorem has many applications in real-world problems, such as modeling the number of phone calls received by a call center, the number of customers arriving at a store, or the number of defects in a manufacturing process. In each of these cases, the Poisson distribution can be used to model the number of events occurring in a fixed interval of time or space.

Q: What are some common misconceptions about the Poisson Limit Theorem?

A: Some common misconceptions about the Poisson Limit Theorem include:

• The Poisson Limit Theorem only applies to large values of $n$.
• The Poisson distribution is only used to model rare events.
• The Poisson Limit Theorem is only applicable to discrete distributions.

These misconceptions are not true, and the Poisson Limit Theorem can be applied to a wide range of problems, including both large and small values of $n$, and both discrete and continuous distributions.

Conclusion

In conclusion, the Poisson Limit Theorem is a fundamental concept in probability theory that describes the limiting behavior of the binomial distribution as the number of trials increases. The parameter $\lambda$ is calculated as $\lambda = np$, where $n$ is the number of trials and $p$ is the probability of success. The Poisson distribution is a continuous distribution that models the number of events occurring in a fixed interval of time or space. The Poisson Limit Theorem has many applications in real-world problems, and it is an important tool for modeling and analyzing complex systems.

Frequently Asked Questions

Q: What is the Poisson Limit Theorem?

A: The Poisson Limit Theorem is a result that describes the limiting behavior of the binomial distribution as the number of trials increases.

Q: What is the parameter $\lambda$ in the Poisson distribution?

A: The parameter $\lambda$ represents the average rate of events occurring in a fixed interval of time or space.

Q: How is the parameter $\lambda$ calculated?

A: The parameter $\lambda$ is calculated as $\lambda = np$, where $n$ is the number of trials and $p$ is the probability of success.

Q: What is the relationship between the binomial distribution and the Poisson distribution?

A: The binomial distribution can be viewed as a sequence of independent Bernoulli trials, each with a probability of success $p$. The number of successes in $n$ trials is then a binomial random variable with parameters $n$ and $p$. As $n$ increases, the binomial distribution converges to the Poisson distribution with parameter $\lambda = np$.

References

• [1] Feller, W. (1968). An Introduction to Probability Theory and Its Applications. Wiley.
• [2] Grimmett, G. R., & Stirzaker, D. R. (2001). Probability and Random Processes. Oxford University Press.
• [3] Ross, S. M. (2010). Introduction to Probability Models. Academic Press.

Further Reading

• [1] Poisson Distribution on Wikipedia
• [2] Binomial Distribution on Wikipedia
• [3] Poisson Limit Theorem on MathWorld