Poisson as a Model and Approximation to Binomial

Introduction

Key Concepts

1. Understanding the Binomial Distribution

The binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of independent trials, each with the same probability of success. It is defined by two parameters: the number of trials ($n$) and the probability of success ($p$) in a single trial.

The probability mass function (PMF) of a binomial distribution is given by: $$ P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k} $$ where $\binom{n}{k}$ is the binomial coefficient, representing the number of ways to choose $k$ successes out of $n$ trials.

**Example:** Consider flipping a fair coin ($p = 0.5$) 10 times ($n = 10$). The probability of getting exactly 4 heads is: $$ P(X = 4) = \binom{10}{4} (0.5)^4 (0.5)^6 = 210 \times 0.0625 \times 0.015625 = 0.205 $$

2. Introduction to the Poisson Distribution

The Poisson distribution is another discrete probability distribution that models the number of events occurring within a fixed interval of time or space. Unlike the binomial distribution, it is characterized by a single parameter, $\lambda$, which represents the average rate of occurrence.

The PMF of the Poisson distribution is: $$ P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!} $$ where $k$ is the number of occurrences, and $e$ is the base of the natural logarithm.

**Example:** If a website receives an average of 3 visitors per hour ($\lambda = 3$), the probability of receiving exactly 5 visitors in the next hour is: $$ P(X = 5) = \frac{3^5 e^{-3}}{5!} = \frac{243 \times 0.0498}{120} \approx 0.1008 $$

3. Conditions for Poisson Approximation to the Binomial

The Poisson distribution can approximate the binomial distribution under specific conditions:

• Large Number of Trials: The number of trials ($n$) is large.
• Small Probability of Success: The probability of success ($p$) in each trial is small.
• Constant Mean: The product $np = \lambda$ remains moderate.

When these conditions are met, the binomial distribution $B(n, p)$ can be approximated by Poisson distribution $Po(\lambda)$, where $\lambda = np$.

**Rationale:** As $n$ increases and $p$ decreases while keeping $\lambda = np$ constant, the binomial distribution becomes increasingly similar to the Poisson distribution. This is particularly useful because the Poisson distribution often simplifies calculations in such scenarios.

4. Deriving the Poisson Approximation

To derive the Poisson approximation to the binomial distribution, consider the binomial PMF: $$ P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k} $$ As $n \to \infty$ and $p \to 0$ such that $\lambda = np$ remains constant, the binomial coefficient can be approximated using Stirling's formula: $$ \binom{n}{k} \approx \frac{n^k}{k!} $$ Substituting $p = \frac{\lambda}{n}$: $$ P(X = k) \approx \frac{n^k}{k!} \left(\frac{\lambda}{n}\right)^k \left(1 - \frac{\lambda}{n}\right)^{n - k} $$ Simplifying: $$ P(X = k) \approx \frac{\lambda^k}{k!} \left(1 - \frac{\lambda}{n}\right)^n \left(1 - \frac{\lambda}{n}\right)^{-k} $$ As $n \to \infty$, $\left(1 - \frac{\lambda}{n}\right)^n \to e^{-\lambda}$ and $\left(1 - \frac{\lambda}{n}\right)^{-k} \to 1$. Therefore: $$ P(X = k) \approx \frac{\lambda^k e^{-\lambda}}{k!} $$ which is the PMF of the Poisson distribution.

5. Examples of Poisson Approximation

Let's explore practical examples where the Poisson approximation to the binomial distribution is applicable:

1. Rare Events in a Large Population: Suppose a factory produces 10,000 light bulbs ($n = 10,000$), and the probability of a defective bulb is 0.001 ($p = 0.001$). Here, $\lambda = np = 10$. The probability of finding exactly 8 defective bulbs can be approximated using the Poisson distribution: $$ P(X = 8) = \frac{10^8 e^{-10}}{8!} \approx 0.1126 $$
2. Customer Arrivals: If a store averages 5 customers per hour ($\lambda = 5$), the probability of exactly 3 customers arriving in an hour is: $$ P(X = 3) = \frac{5^3 e^{-5}}{3!} \approx 0.1404 $$
3. Call Center Operations: A call center receives an average of 12 calls per hour ($\lambda = 12$). The probability of receiving exactly 15 calls in an hour is: $$ P(X = 15) = \frac{12^{15} e^{-12}}{15!} \approx 0.0519 $$

6. Properties of the Poisson Distribution

• Mean and Variance: Both the mean and variance of a Poisson distribution are equal to $\lambda$.
• Skewness: The distribution is skewed to the right, especially for small values of $\lambda$.
• Additivity: The sum of independent Poisson random variables is also Poisson distributed, with parameter equal to the sum of their parameters.

7. Limitations of Poisson Approximation

While the Poisson approximation simplifies calculations, it has limitations:

• Accuracy: The approximation is less accurate when $p$ is not sufficiently small or $n$ is not sufficiently large.
• Discrete Nature: It may not be suitable for modeling events that are not rare or for distributions where the variance significantly deviates from the mean.

8. Applications in Real-World Scenarios

The Poisson approximation is extensively used in various fields:

• Telecommunications: Modeling the number of phone calls received by a call center.
• Biology: Counting the number of mutations in a given stretch of DNA.
• Finance: Estimating the number of defaults in a large portfolio of loans.
• Transportation: Predicting the number of accidents at a particular location over a period.

Advanced Concepts

1. Mathematical Derivation of Poisson Limit Theorem

The Poisson Limit Theorem formalizes the conditions under which a binomial distribution converges to a Poisson distribution. Consider a sequence of binomial distributions $B(n, p_n)$ where $n \to \infty$, $p_n \to 0$, and $np_n \to \lambda$. The theorem states: $$ \lim_{n \to \infty} P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!} $$ To prove this, start with the binomial PMF: $$ P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k} $$ Using the approximation $\binom{n}{k} \approx \frac{n^k}{k!}$ for large $n$ and fixed $k$, and $p_n = \frac{\lambda}{n}$: $$ P(X = k) \approx \frac{n^k}{k!} \left(\frac{\lambda}{n}\right)^k \left(1 - \frac{\lambda}{n}\right)^n = \frac{\lambda^k}{k!} \left(1 - \frac{\lambda}{n}\right)^n $$ Taking the limit as $n \to \infty$: $$ \lim_{n \to \infty} \left(1 - \frac{\lambda}{n}\right)^n = e^{-\lambda} $$ Thus: $$ \lim_{n \to \infty} P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!} $$

2. Generating Functions

Generating functions are powerful tools for analyzing probability distributions. The generating function of the Poisson distribution is: $$ G_X(t) = e^{\lambda(t - 1)} $$ For the binomial distribution $B(n, p)$, the generating function is: $$ G_X(t) = [1 - p + pt]^n $$> To derive the Poisson generating function from the binomial one, set $p = \frac{\lambda}{n}$ and take the limit as $n \to \infty$: $$ G_X(t) = \left(1 - \frac{\lambda}{n} + \frac{\lambda}{n} t\right)^n \approx \left(1 + \frac{\lambda(t - 1)}{n}\right)^n \to e^{\lambda(t - 1)} $$> Thus, the generating functions converge, reinforcing the Poisson approximation.

3. Confidence Intervals and Hypothesis Testing

Understanding the Poisson approximation facilitates constructing confidence intervals and conducting hypothesis tests for count data. For instance, estimating the rate parameter $\lambda$ can be approached using interval estimates derived from the Poisson distribution properties. **Example:** To construct a 95% confidence interval for $\lambda$ based on observed data $k$, we can use the following approximation: $$ \lambda \approx k \pm 1.96\sqrt{k} $$ where 1.96 corresponds to the z-score for a 95% confidence level.

4. Interdisciplinary Connections

The Poisson distribution's applicability extends to various disciplines:

• Physics: Modeling radioactive decay events over time.
• Environmental Science: Predicting the occurrence of natural disasters like earthquakes.
• Medicine: Estimating the number of disease occurrences in a population.
• Economics: Analyzing rare economic events, such as market crashes.

These connections underscore the Poisson distribution's versatility and its foundational role in statistical modeling across diverse fields.

5. Advanced Problem-Solving Techniques

Applying the Poisson approximation requires a deep understanding of both the binomial and Poisson distributions. Here's a multi-step problem to illustrate this:

Problem: A bookstore receives an average of 2 orders per day for a rare book. What is the probability that exactly 5 orders are received in a week? Assume orders are independent and occur with a constant average rate.

Solution:

1. Determine Parameters: The average rate per day ($\lambda_d$) is 2 orders. For a week ($7$ days), $\lambda_w = 2 \times 7 = 14$.
2. Apply Poisson PMF: $$ P(X = 5) = \frac{14^5 e^{-14}}{5!} $$
3. Calculate: $$ 14^5 = 537,824 \\ 5! = 120 \\ e^{-14} \approx 8.3153 \times 10^{-7} \\ P(X = 5) = \frac{537,824 \times 8.3153 \times 10^{-7}}{120} \approx 0.0372 $$

Thus, the probability of receiving exactly 5 orders in a week is approximately 3.72%.

Comparison Table

Summary and Key Takeaways