Poisson Binomial Approximation

16.6 Poisson Distribution as an Approximation to the Binomial Distribution

The Poisson distribution serves as a practical and often simpler approximation to the Binomial distribution under specific conditions. The Binomial distribution is typically used to calculate the probability of a certain number of successes in a fixed number of independent trials. However, when the number of trials ($n$) is large and the probability of success ($p$) is small, calculating Binomial probabilities can become computationally intensive and cumbersome.

In such scenarios, the Poisson distribution offers a more manageable alternative.

When to Use Poisson Approximation

The Poisson approximation to the Binomial distribution is particularly suitable when the following conditions are met:

• Large number of trials ($n$): Generally, $n \ge 30$.

• Small probability of success ($p$): Typically, $p \le 0.05$.

• Moderate average rate of success ($\lambda$): The product of $n$ and $p$, denoted as $\lambda = np$, should be a moderate value (often considered to be less than 10).

This approximation technique is especially valuable for modeling rare events that occur over a fixed interval of time or space.

Poisson Distribution Formula

The probability mass function (PMF) of the Poisson distribution is given by:

$$ P(X = x) = \frac{e^{-\lambda} \lambda^x}{x!} $$

Where:

• $P(X = x)$: The probability of exactly $x$ occurrences.

• $\lambda$ ($lambda$): The average number of occurrences in the given interval. In the context of the Binomial approximation, $\lambda = np$.

• $x$: The actual number of observed occurrences.

• $e$: The base of the natural logarithm, approximately $2.71828$.

• $x!$: The factorial of $x$ ($x \times (x-1) \times \dots \times 1$).

Example: Approximating Binomial with Poisson

Consider a scenario in a telecom service center where only 1% of daily service tickets are found to have processing errors. If 250 tickets are reviewed in a day, what is the probability that exactly 3 tickets will have processing errors? We will use the Poisson distribution to approximate this Binomial probability.

Step 1: Identify Parameters and Check Conditions

First, we identify the parameters of the Binomial distribution and check if the conditions for Poisson approximation are met:

• Number of trials ($n$): $n = 250$

• Probability of success ($p$): $p = 0.01$

Conditions Check:

• $n = 250 \ge 30$ (Large number of trials - Met)

• $p = 0.01 \le 0.05$ (Small probability of success - Met)

Now, we calculate the average rate ($\lambda$) for the Poisson distribution:

• Average rate ($\lambda$): $\lambda = np = 250 \times 0.01 = 2.5$ Since $\lambda = 2.5$ is moderate (less than 10), the Poisson approximation is appropriate.

Step 2: Use the Poisson Formula

We want to find the probability of exactly 3 processing errors ($x=3$). Using the Poisson formula:

$$ P(X = 3) = \frac{e^{-2.5} (2.5)^3}{3!} $$

Step-by-step Calculation:

1. Calculate $e^{-2.5}$: $e^{-2.5} \approx 0.082085$

2. Calculate $(2.5)^3$: $(2.5)^3 = 15.625$

3. Calculate $3!$: $3! = 3 \times 2 \times 1 = 6$

4. Substitute these values into the formula: $$ P(X = 3) = \frac{0.082085 \times 15.625}{6} $$ $$ P(X = 3) = \frac{1.282578125}{6} $$ $$ P(X = 3) \approx 0.21376 $$

Final Answer

The probability that exactly 3 out of 250 service tickets have processing errors, using the Poisson distribution as an approximation, is approximately 0.214 or 21.4%.

Summary

The Poisson approximation to the Binomial distribution is a valuable technique for simplifying probability calculations involving rare events in large sample sizes. By transforming the Binomial problem into a Poisson one using $\lambda = np$, we can effectively manage complex calculations and obtain accurate probability estimates with a simpler model.

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Interview Questions

• When is it appropriate to use the Poisson distribution as an approximation to the Binomial distribution? It's appropriate when the number of trials ($n$) is large (e.g., $n \ge 30$) and the probability of success ($p$) is small (e.g., $p \le 0.05$), provided that the product $\lambda = np$ remains moderate.

• What are the key assumptions for using Poisson as an approximation? The underlying process should be a sequence of Bernoulli trials, but the large $n$ and small $p$ conditions allow us to approximate it with a Poisson process where events occur independently at a constant average rate.

• How do you derive $\lambda$ from Binomial parameters in Poisson approximation? $\lambda$ is derived by multiplying the number of trials ($n$) by the probability of success ($p$), i.e., $\lambda = np$.

• Why does the Poisson distribution work well for rare events? For rare events, the probability of multiple events occurring in a small interval is very low. The Poisson distribution inherently models the number of events in a fixed interval, making it a good fit for situations where events are infrequent but occur over a large number of opportunities.

• Can you explain the difference between Binomial and Poisson distributions? The Binomial distribution deals with a fixed number of trials, each with two outcomes (success/failure), and calculates the probability of a specific number of successes. The Poisson distribution models the number of events occurring in a fixed interval of time or space, given a constant average rate, without a fixed number of trials.

• What is the Poisson formula and what does each term represent? The formula is $P(X = x) = \frac{e^{-\lambda} \lambda^x}{x!}$. $P(X=x)$ is the probability of exactly $x$ events, $\lambda$ is the average rate of events, $x$ is the observed number of events, $e$ is Euler's number, and $x!$ is the factorial of $x$.

• In the given telecom example, why is Poisson a good approximation? In the telecom example, $n=250$ (large) and $p=0.01$ (small), and $\lambda = np = 2.5$ (moderate). These conditions perfectly align with the criteria for using the Poisson approximation, making it a suitable and simpler alternative to calculating the exact Binomial probability.

• How do you interpret the result of a Poisson probability in a real-world context? The result, like $0.214$ in the example, means there is approximately a 21.4% chance of observing exactly 3 processing errors among the 250 tickets reviewed, given the known average error rate.

• What are some real-world examples where Poisson approximation is useful? Counting the number of defects in a manufactured item, the number of customers arriving at a service desk per hour, the number of accidents at an intersection per month, or the number of emails received per minute.

• What happens to the accuracy of the Poisson approximation if $p$ increases or $n$ decreases? If $p$ increases (and $n$ stays large), the approximation becomes less accurate because the assumption of "rare" events is violated. Similarly, if $n$ decreases (and $p$ stays small), the number of trials is no longer sufficiently large for the approximation to hold well. The accuracy is best when both $n$ is large and $p$ is small.