Question

Show that, as the number of trials \(n\) becomes large but \(n p_{i}=\lambda_{i}, i=1,2, \ldots, k-1\) remains finite, the multinomial probability distribution (26.146), $$ M_{n}\left(x_{1}, x_{2}, \ldots, x_{k}\right)=\frac{n !}{x_{1} ! x_{2} ! \cdots x_{k} !} p_{1}^{x_{1}} p_{2}^{x_{2}} \cdots p_{k}^{x_{k}} $$ can be approximated by a multiple Poisson distribution (with \(k-1\) factors) $$ M_{n}^{\prime}\left(x_{1}, x_{2}, \ldots, x_{k-1}\right)=\prod_{i=1}^{k-1} \frac{e^{-\lambda_{i}} \lambda_{i}^{x_{i}}}{x_{i} !} $$ (Write \(\sum_{i}^{k-1} p_{i}=\delta\) and express all terms involving subscript \(k\) in terms of \(n\) and \(\delta\), either exactly or approximately. You will need to use \(n ! \approx n^{f}[(n-\epsilon) !]\) and \((1-a / n)^{n} \approx e^{-a}\) for large \(\left.n_{1}\right)\) (a) Verify that the terms of \(M_{n}^{\prime}\) when summed over all values of \(x_{1}, x_{2}, \ldots, x_{k-1}\) add up to unity. (b) If \(k=7\) and \(\lambda_{i}=9\) for all \(i=1,2, \ldots, 6\), estimate, using the appropriate Gaussian approximation, the chance that at least three of \(x_{1}, x_{2}, \ldots, x_{6}\) will be 15 or greater.

Short answer

The multinomial distribution approximates to the multiple Poisson distribution as \(n \to \infty\) and \(n p_i = \lambda_i\). Summed terms add to 1. At least three values \( \geq 15 \) probability can be calculated using Gaussian approximation for given \( \lambda_i\).

Step-by-step solution

Understand the Multinomial Distribution

The multinomial distribution represents the probability of obtaining a specific number of outcomes in different categories over multiple trials. The formula provided is \[ M_{n}(x_{1}, x_{2}, \ldots, x_{k}) = \frac{n!}{x_{1}! x_{2}! \cdots x_{k}!} p_{1}^{x_{1}} p_{2}^{x_{2}} \cdots p_{k}^{x_{k}} \] where \( p_{i} \) represents the probability of the i-th category, and \( x_{i} \) represents the occurrence count in that category.

Apply Conditions

Given that \(n p_{i} = \lambda_{i} \) remains finite, we convert the terms involving subscript \( k \) to express everything in terms of \( n \) and \( \delta \). Here \( \delta = \sum_{i}^{k-1} p_{i} \) and the remaining probability for the k-th category is approximately \( 1- \delta \).

Approximate Factorials using Stirling's Approximation

\( n! \approx n^n e^{-n} \sqrt{2 \pi n} \), for large \( n \). This approximation eases the calculations when \( n \) is very large.

Approximate Multinomial Probability Distribution

Using the provided approximations: \( (1-a / n)^{n} \approx e^{-a} \) for large \( n \). Transform the multinomial expression to a Poisson form: \[ \prod_{i=1}^{k-1} \frac{e^{-\lambda_{i}} \lambda_{i}^{x_{i}}}{x_{i}!} \].

Verify Sum to Unity

Show that when summed over all possible values of \( x_{1}, x_{2}, \ldots, x_{k-1}, \), the multiple Poisson distribution equals 1. Utilize the identity for Poisson distributions summing to 1.

Compute for Given Parameters

With \( k =7 \), \( \lambda_{i} = 9 \), and using Gaussian approximation, calculate the probability that at least three of \( x_{1}, x_{2}, \ldots, x_{6} \) are 15 or greater.

Key concepts

Poisson Distribution

The Poisson distribution is a powerful probability distribution used to model the number of events occurring within a fixed interval of time or space. These events are considered to happen independently of each other and at a constant rate. For example, the number of customer arrivals at a bank in an hour can be modeled using a Poisson distribution. The key parameters here are:

• \textbf{Rate parameter} (\textlambda): The average number of events in the interval.

The Poisson distribution formula is given by: \[ P(X = k) = \frac{\textlambda^k e^{- \textlambda}}{k!} \] where:
\begin{itemize}

\textlambda is the average rate of occurrence.

k is the actual number of events.

e is the base of the natural logarithm (approximately 2.71828).

This distribution directly applies to modeling the multinomial distribution as a multiple Poisson distribution, provided the number of trials (n) is large and the product of the number of trials and the probability of occurrence (\textlambda) remains finite.

Stirling's Approximation

Stirling's Approximation is a mathematical tool used to approximate factorials, especially helpful when dealing with large numbers. This approximation simplifies the computation significantly. The formula for Stirling's Approximation is:
\[ n! \approx n^n e^{-n} \sqrt{2\textpi\ n} \] where:

• n! denotes the factorial of n.
• e is the base of the natural logarithm.
• \textpi\ is the constant pi (approximately 3.14159).

In the context of the multinomial distribution problem, Stirling's Approximation helps convert factorial terms involved in large n calculations into a more manageable form. For example, instead of calculating an extremely large factorial directly, we use the approximation for more straightforward computation. This technique is crucial in transforming the multinomial expression into a multiple Poisson form by simplifying the n! and k! terms. Thus, Stirling's Approximation streamlines the transition from a complex multinomial distribution to a more tractable Poisson distribution.

Gaussian Approximation

The Gaussian Approximation, also known as the Normal Approximation, is used to approximate the binomial or Poisson distributions when the number of trials is large. The presence of \textlambda being large facilitates this approximation, which simplifies the problem-solving process. The Gaussian (Normal) distribution has two parameters: the mean (\textmu) and the variance (\textsigma^2).
The Central Limit Theorem states that as the number of trials (n) becomes large, the distribution of the sum (or average) of the trials will approach a normal distribution. For a Poisson distribution with rate \textlambda, a Gaussian approximation would mean:

• \textmu = \textlambda
• \textsigma^2 = \textlambda

In practical terms, if we need to find the probability of at least three of \text{x_1, x_2, ..., x_6} being 15 or greater given \textlambda\text_i = 9, we use the Gaussian distribution to approximate this by standardizing the required outcomes. The formula for standardizing an observation (z-score) is: \[ z = \frac{X - \textmu} {\textsigma} \] This approach converts the problem into finding probabilities associated with the standard normal distribution, making it easier to handle. The Gaussian Approximation thus facilitates solving problems that would otherwise involve cumbersome Poisson probabilities.