Poisson Approximation to the Binomial

Poisson Approximation to the Binomial

The Poisson distribution is a powerful tool for approximating the binomial distribution when certain conditions are met:

• The number of trials \(n\) is large
• The success probability \(p\) is small
• The expected number of successes \(np\) is moderate, typically \(np < 5\)

In such cases: \[ \text{Binomial}(n, p) \approx \text{Poisson}(\lambda = np) \]

This approximation greatly simplifies probability calculations—especially when exact binomial formulas are unwieldy.

Tip: If \(p\) is high, redefine “success” and “failure” to make \(p\) small.

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Example 1: Comparing Binomial and Poisson

Imagine inspecting 1000 items from a production line with a 0.1% defect rate.

Using the Binomial Formula

We want to compute the probability of finding exactly 3 defective items:

\[ P(X = 3) = \binom{1000}{3} \cdot (0.001)^3 \cdot (0.999)^{997} \]

Breaking it down: - \(\binom{1000}{3} = 166{,}167{,}000\) - \((0.001)^3 = 1 \times 10^{-9}\) - \(0.999^{997} \approx 0.36880\)

\[ P(X = 3) \approx 166{,}167{,}000 \cdot 10^{-9} \cdot 0.36880 \approx 0.06128 \]

Using the Poisson Approximation

Check the criteria: - \(n = 1000\) - \(np = 1000 \times 0.001 = 1 < 5\)

Now apply the Poisson formula:

\[ P(X = 3) = \frac{e^{-1} \cdot 1^3}{3!} = \frac{0.36787}{6} \approx 0.06131 \]

Comparison

Method	Result
Binomial	0.06128
Poisson	0.06131

Very close—the Poisson method is much faster and still accurate.

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Example 2: Cumulative Probability

Use the Poisson approximation to compute: \[ P(X \geq 2) \]

From the Poisson distribution with \(\lambda = 2\): \[ P(X \geq 2) = 1 - P(0) - P(1) \] \[ = 1 - e^{-2} - 2e^{-2} = 1 - 3e^{-2} \approx 1 - 0.406 = 0.594 \]

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Refresher: The Poisson Distribution

Probability Density Function (PDF):

\[ P(X = k) = \frac{e^{-\lambda} \cdot \lambda^k}{k!} \]

Where: - \(\lambda\) is the expected number of events - \(k\) is the number of actual occurrences

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Example: Modeling Arrivals

A system receives 15 arrivals per half hour on average. What’s the probability no arrivals occur in a 5-minute window?

5 minutes is 1/6 of 30 minutes, so: \[ \lambda = \frac{1}{6} \cdot 15 = 2.5 \] \[ P(X = 0) = e^{-2.5} \approx 0.0821 \]

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Summary: When to Use Poisson Approximation

• \(n \geq 50\)
• \(p \leq 0.1\)
• \(np \leq 5\)
• Especially useful for computing binomial probabilities for small values of \(X\)

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