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, used to describe the behavior of discrete random variables as the number of trials increases. However, when applying this theorem, it's not uncommon to encounter confusion, especially when dealing with specific scenarios. In this article, we'll delve into a particular application of the Poisson Limit Theorem, where the parameter $\lambda$ is expressed as $\frac{m}{n}$. We'll explore the reasoning behind this expression and provide a step-by-step explanation to clarify the concept.

Background on the Poisson Limit Theorem

The Poisson Limit Theorem states that if we have a sequence of independent and identically distributed (i.i.d.) random variables $X_1, X_2, \ldots, X_m$, each with a Poisson distribution with parameter $\lambda$, then as $m$ approaches infinity, the distribution of the sum of these random variables converges to a Poisson distribution with parameter $\lambda$. Mathematically, this can be expressed as:

$$\lim_{m \to \infty} \Pr\left(\sum_{i=1}^m X_i = r\right) = \frac{\mathrm{e}^{-\lambda} \lambda^r}{r!}$$

The Confusing Application

The application in question involves a sequence of random variables $X_m$, where each $X_m$ represents the number of successes in $m$ independent trials, each with a probability of success $p$. The probability mass function (PMF) of $X_m$ is given by:

$$\Pr(X_m = r) = \binom{m}{r} p^r (1-p)^{m-r}$$

The goal is to show that as $m$ approaches infinity, the distribution of $X_m$ converges to a Poisson distribution with parameter $\lambda = \frac{m}{n}$.

The Reasoning Behind $\lambda = \frac{m}{n}$

To understand why $\lambda = \frac{m}{n}$, let's consider the following:

• The parameter $\lambda$ represents the expected value of the Poisson distribution, which is equal to the average number of successes in a fixed number of trials.
• In this scenario, we have $m$ independent trials, each with a probability of success $p$.
• The expected value of $X_m$ is given by $E(X_m) = mp$, which represents the average number of successes in $m$ trials.
• To obtain a Poisson distribution with parameter $\lambda = \frac{m}{n}$, we need to scale the expected value of $X_m$ by a factor of $\frac{1}{n}$.

Derivation of the Limit

To derive the limit, we'll use the following steps:

1. Apply the Central Limit Theorem (CLT): The CLT states that the distribution of the sum of i.i.d. random variables converges to a normal distribution as the number of variables increases. In this case, we can apply the CLT to the sequence of random variables $X_m$.
2. Use the Poisson approximation to the normal distribution: As the number of trials increases, the normal distribution can beimated by a Poisson distribution with parameter $\lambda = \frac{m}{n}$.
3. Derive the limit: Using the Poisson approximation, we can derive the limit of the probability mass function of $X_m$ as $m$ approaches infinity.

Derivation of the Limit (continued)

Let's continue with the derivation:

1. Apply the CLT: The CLT states that the distribution of the sum of i.i.d. random variables converges to a normal distribution as the number of variables increases. In this case, we can apply the CLT to the sequence of random variables $X_m$.

$$\lim_{m \to \infty} \Pr\left(\sum_{i=1}^m X_i = r\right) = \frac{1}{\sqrt{2\pi \sigma^2}} \int_{-\infty}^r \mathrm{e}^{-\frac{(x-\mu)^2}{2\sigma^2}} \mathrm{d}x$$

where $\mu = mp$ and $\sigma^2 = mp(1-p)$.

2. Use the Poisson approximation to the normal distribution: As the number of trials increases, the normal distribution can be approximated by a Poisson distribution with parameter $\lambda = \frac{m}{n}$.

$$\lim_{m \to \infty} \Pr\left(\sum_{i=1}^m X_i = r\right) = \frac{\mathrm{e}^{-\lambda} \lambda^r}{r!}$$

where $\lambda = \frac{m}{n}$.

3. Derive the limit: Using the Poisson approximation, we can derive the limit of the probability mass function of $X_m$ as $m$ approaches infinity.

$$\lim_{m \to \infty} \Pr(X_m = r) = \frac{\mathrm{e}^{-m/n} (m/n)^r}{r!}$$

Conclusion

In this article, we've explored a particular application of the Poisson Limit Theorem, where the parameter $\lambda$ is expressed as $\frac{m}{n}$. We've provided a step-by-step explanation of the reasoning behind this expression and derived the limit of the probability mass function of $X_m$ as $m$ approaches infinity. The result shows that the distribution of $X_m$ converges to a Poisson distribution with parameter $\lambda = \frac{m}{n}$.

References

• [1] Feller, W. (1968). An Introduction to Probability Theory and Its Applications. John Wiley & Sons.
• [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.

Additional Resources

• [1] Poisson Distribution: A Tutorial by Khan Academy
• [2] Poisson Limit Theorem: A Proof by Math Stack Exchange
• [3] Central Limit Theorem: A Proof by MIT OpenCourseWare
  

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Introduction

In our previous article, we explored a particular application of the Poisson Limit Theorem, where the parameter $\lambda$ is expressed as $\frac{m}{n}$. We provided a step-by-step explanation of the reasoning behind this expression and derived the limit of the probability mass function of $X_m$ as $m$ approaches infinity. However, we understand that some readers may still have questions or concerns about this topic. In this article, we'll address some of the most frequently asked questions (FAQs) related to this application of the Poisson Limit Theorem.

Q&A

Q1: What is the Poisson Limit Theorem, and why is it important?

A1: The Poisson Limit Theorem is a fundamental concept in probability theory that describes the behavior of discrete random variables as the number of trials increases. It's essential in understanding the distribution of rare events, such as the number of defects in a manufacturing process or the number of accidents in a given time period.

Q2: How does the Poisson Limit Theorem relate to the Poisson distribution?

A2: The Poisson Limit Theorem states that the distribution of the sum of independent and identically distributed (i.i.d.) random variables converges to a Poisson distribution as the number of variables increases. This means that the Poisson distribution can be used to approximate the distribution of the sum of i.i.d. random variables.

Q3: What is the significance of $\lambda = \frac{m}{n}$ in the Poisson Limit Theorem?

A3: The parameter $\lambda$ represents the expected value of the Poisson distribution, which is equal to the average number of successes in a fixed number of trials. In this scenario, $\lambda = \frac{m}{n}$ represents the average number of successes in $m$ trials, scaled by a factor of $\frac{1}{n}$.

Q4: How do I apply the Poisson Limit Theorem in real-world scenarios?

A4: The Poisson Limit Theorem can be applied in various real-world scenarios, such as:

• Modeling the number of defects in a manufacturing process
• Estimating the number of accidents in a given time period
• Analyzing the number of customers arriving at a store or website
• Predicting the number of sales or revenue in a given time period

Q5: What are some common mistakes to avoid when applying the Poisson Limit Theorem?

A5: Some common mistakes to avoid when applying the Poisson Limit Theorem include:

• Failing to check the conditions for the Poisson Limit Theorem, such as independence and identical distribution
• Using an incorrect value for $\lambda$
• Failing to account for the scaling factor $\frac{1}{n}$

Q6: Can the Poisson Limit Theorem be applied to continuous random variables?

A6: The Poisson Limit Theorem is typically applied to discrete random variables. However, it can be extended to continuous random variables using the concept of the Poisson process.

Q7: What are some real-world applications of the Poisson Limit Theorem?

A7: Some real-world applications of the Poisson Limit Theorem include* Modeling the number of defects in a manufacturing process

• Estimating the number of accidents in a given time period
• Analyzing the number of customers arriving at a store or website
• Predicting the number of sales or revenue in a given time period

Conclusion

In this article, we've addressed some of the most frequently asked questions (FAQs) related to the application of the Poisson Limit Theorem. We hope that this article has provided a better understanding of this fundamental concept in probability theory and its real-world applications.

References

• [1] Feller, W. (1968). An Introduction to Probability Theory and Its Applications. John Wiley & Sons.
• [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.

Additional Resources

• [1] Poisson Distribution: A Tutorial by Khan Academy
• [2] Poisson Limit Theorem: A Proof by Math Stack Exchange
• [3] Central Limit Theorem: A Proof by MIT OpenCourseWare