How To Prove NegativeBinomial ( R , P ) \text{NegativeBinomial}(r,p) NegativeBinomial ( R , P ) Converges To Gamma ( R , 1 ) \text{Gamma}(r,1) Gamma ( R , 1 ) As P → 0 P \to 0 P → 0 ?

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Introduction

The Negative Binomial Distribution and the Gamma Distribution are two distinct probability distributions used in various fields of statistics and probability theory. The Negative Binomial Distribution is often used to model the number of failures until a specified number of successes is achieved, while the Gamma Distribution is used to model the time until a specified event occurs. In this article, we will explore the relationship between these two distributions and prove that the Negative Binomial Distribution converges to the Gamma Distribution as the probability of success $p$ approaches 0.

Background

The Negative Binomial Distribution is a discrete probability distribution that models the number of failures until a specified number of successes is achieved. It is characterized by two parameters: the number of successes $r$ and the probability of success $p$. The probability mass function of the Negative Binomial Distribution is given by:

$$P(X=k) = \binom{k+r-1}{r-1} p^r (1-p)^k$$

where $k$ is the number of failures.

The Gamma Distribution, on the other hand, is a continuous probability distribution that models the time until a specified event occurs. It is characterized by two parameters: the shape parameter $\alpha$ and the rate parameter $\beta$. The probability density function of the Gamma Distribution is given by:

$$f(x) = \frac{\beta^\alpha}{\Gamma(\alpha)} x^{\alpha-1} e^{-\beta x}$$

where $x$ is the time until the event occurs.

Convergence of the Negative Binomial Distribution to the Gamma Distribution

To prove that the Negative Binomial Distribution converges to the Gamma Distribution as $p$ approaches 0, we need to show that the probability mass function of the Negative Binomial Distribution converges to the probability density function of the Gamma Distribution.

Let $X\sim \text{NegBin}(r,p)$ and $Y\sim \text{Gamma}(r,1)$. We want to show that $pX \overset{\text{dist}}\longrightarrow Y$ as $p\to 0$.

Proof

To prove this statement, we can use the following approach:

1. Show that the probability mass function of the Negative Binomial Distribution converges to the probability density function of the Gamma Distribution as $p$ approaches 0.
2. Use the definition of convergence in distribution to show that $pX \overset{\text{dist}}\longrightarrow Y$ as $p\to 0$.

Step 1: Convergence of the Probability Mass Function

We start by showing that the probability mass function of the Negative Binomial Distribution converges to the probability density function of the Gamma Distribution as $p$ approaches 0.

Let $k$ be a positive integer. Then, we have:

$$P(pX=k) = P(X=k/p) = \binom{k/p+r-1}{r-1} p^r (1-p)^{k/p}$$

As $p$ approaches 0, we have:

$$\lim_{p\to 0} P(pX=k) = \lim_{p\to 0} \binom{k/p+r-1}{r-1} p^r (1-p)^{k}$$

Using the fact that $\lim_{x\to 0} (1+x)^{1/x} = e$, we have:

$$\lim_{p\to 0} P(pX=k) = \lim_{p\to 0} \binom{k/p+r-1}{r-1} p^r (1-p)^{k/p} = \frac{\Gamma(k+r)}{\Gamma(r)} \frac{1}{k^r}$$

which is the probability density function of the Gamma Distribution with shape parameter $r$ and rate parameter 1.

Step 2: Convergence in Distribution

Now, we use the definition of convergence in distribution to show that $pX \overset{\text{dist}}\longrightarrow Y$ as $p\to 0$.

Let $F_p$ be the cumulative distribution function of $pX$ and $F$ be the cumulative distribution function of $Y$. Then, we have:

$$F_p(x) = P(pX\leq x) = \sum_{k=0}^{\lfloor x/p \rfloor} P(pX=k)$$

As $p$ approaches 0, we have:

$$\lim_{p\to 0} F_p(x) = \lim_{p\to 0} \sum_{k=0}^{\lfloor x/p \rfloor} P(pX=k) = \int_0^x \frac{\Gamma(k+r)}{\Gamma(r)} \frac{1}{k^r} dk$$

which is the cumulative distribution function of the Gamma Distribution with shape parameter $r$ and rate parameter 1.

Therefore, we have shown that $pX \overset{\text{dist}}\longrightarrow Y$ as $p\to 0$.

Conclusion

In this article, we have explored the relationship between the Negative Binomial Distribution and the Gamma Distribution. We have shown that the Negative Binomial Distribution converges to the Gamma Distribution as the probability of success $p$ approaches 0. This result has important implications in various fields of statistics and probability theory, including insurance, finance, and engineering.

References

• Johnson, N. L., Kotz, S., & Kemp, A. W. (1993). Univariate discrete distributions. John Wiley & Sons.
• Evans, M., Hastings, N., & Peacock, B. (2000). Statistical distributions. John Wiley & Sons.
• Feller, W. (1971). An introduction to probability theory and its applications. John Wiley & Sons.
  

Introduction

In our previous article, we explored the relationship between the Negative Binomial Distribution and the Gamma Distribution. We showed that the Negative Binomial Distribution converges to the Gamma Distribution as the probability of success $p$ approaches 0. In this article, we will answer some frequently asked questions about this convergence.

Q: What is the significance of this convergence?

A: The convergence of the Negative Binomial Distribution to the Gamma Distribution has important implications in various fields of statistics and probability theory. For example, it can be used to model the number of failures until a specified number of successes is achieved, where the probability of success is small.

Q: What are the conditions for this convergence to hold?

A: The convergence of the Negative Binomial Distribution to the Gamma Distribution holds when the probability of success $p$ approaches 0. This means that the number of failures until a specified number of successes is achieved must be large.

Q: How can I use this convergence in practice?

A: You can use this convergence to model the number of failures until a specified number of successes is achieved, where the probability of success is small. For example, you can use it to model the number of claims until a specified number of policies is sold.

Q: What are the limitations of this convergence?

A: The convergence of the Negative Binomial Distribution to the Gamma Distribution is only an approximation. It is not exact, and it may not hold in all cases. For example, it may not hold when the number of failures until a specified number of successes is achieved is small.

Q: Can I use this convergence for other distributions?

A: Yes, you can use this convergence for other distributions. For example, you can use it to model the number of failures until a specified number of successes is achieved, where the probability of success is small, and the distribution of the number of failures is not Negative Binomial.

Q: How can I prove this convergence mathematically?

A: You can prove this convergence mathematically by showing that the probability mass function of the Negative Binomial Distribution converges to the probability density function of the Gamma Distribution as the probability of success $p$ approaches 0.

Q: What are the assumptions of this convergence?

A: The assumptions of this convergence are that the probability of success $p$ approaches 0, and the number of failures until a specified number of successes is achieved is large.

Q: Can I use this convergence for Bayesian inference?

A: Yes, you can use this convergence for Bayesian inference. For example, you can use it to model the number of failures until a specified number of successes is achieved, where the probability of success is small, and the distribution of the number of failures is not Negative Binomial.

Q: How can I implement this convergence in R?

A: You can implement this convergence in R by using the dnbinom function to model the Negative Binomial Distribution, and the dgamma function to model the Gamma Distribution.

Q: What are the advantages of this convergence?

A: The advantages of this convergence are that it can be used to model the number of failures until a specified number of successes is achieved, where the probability of success is small and it can be used for Bayesian inference.

Q: What are the disadvantages of this convergence?

A: The disadvantages of this convergence are that it is only an approximation, and it may not hold in all cases.

Conclusion

In this article, we have answered some frequently asked questions about the convergence of the Negative Binomial Distribution to the Gamma Distribution. We have shown that this convergence has important implications in various fields of statistics and probability theory, and it can be used to model the number of failures until a specified number of successes is achieved, where the probability of success is small.

References

• Johnson, N. L., Kotz, S., & Kemp, A. W. (1993). Univariate discrete distributions. John Wiley & Sons.
• Evans, M., Hastings, N., & Peacock, B. (2000). Statistical distributions. John Wiley & Sons.
• Feller, W. (1971). An introduction to probability theory and its applications. John Wiley & Sons.