Normal Approximation to the Poisson Distribution — Mathematics

Introduction

The Poisson distribution is exact and elegant, but when $\lambda$ is large, calculating cumulative probabilities by summing individual Poisson terms becomes extremely tedious — even with tables. The normal distribution provides a powerful and efficient approximation in these cases, turning a discrete probability problem into a continuous one that can be handled with standard normal tables.

For 9709, you must know when this approximation is valid, how to set it up (matching mean and variance), and why a continuity correction is essential whenever you move from a discrete to a continuous distribution.

Core Concept

If $X \sim \text{Po}(\lambda)$ and $\lambda$ is large, then $X$ is approximately normally distributed. Since the Poisson distribution has mean = variance = $\lambda$ , the approximating normal distribution uses both as $\lambda$ :

X \sim \text{Po}(\lambda) \approx Y \sim \text{N}(\lambda,\ \lambda) \quad \text{when } \lambda > 15 \text{ approximately.}

Why does this work? The Poisson distribution can be thought of as a sum of many independent rare-event indicators. By the Central Limit Theorem, the sum of many independent random variables tends towards a normal distribution. As $\lambda$ grows, the Poisson distribution becomes increasingly symmetric and bell-shaped, making the normal approximation progressively more accurate.

The Continuity Correction

Because $X$ is discrete (it takes only integer values) and $Y$ is continuous, you must apply a continuity correction: each integer value $n$ is represented by the interval $(n - 0.5,\ n + 0.5)$ in the continuous normal model.

Poisson probability	Normal approximation (with continuity correction)
$P(X = n)$	$P(n - 0.5 < Y < n + 0.5)$
$P(X \leq n)$	$P(Y < n + 0.5)$
$P(X < n)$	$P(Y < n - 0.5)$
$P(X \geq n)$	$P(Y > n - 0.5)$
$P(X > n)$	$P(Y > n + 0.5)$
$P(a \leq X \leq b)$	$P(a - 0.5 < Y < b + 0.5)$

A helpful mental rule: "expand by 0.5 outwards" — include a little more of the continuous distribution to account for the discrete value being spread across a unit interval.

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Introduction

Core Concept

The Continuity Correction

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