How Does a Poisson Distribution Converge to Normal?

A Poisson random variable takes nonnegative integer values, while a normal random variable is continuous. Even so, a Poisson distribution becomes increasingly bell-shaped as its mean grows. The precise convergence statement concerns a standardized Poisson variable: subtract its mean and divide by its standard deviation.

This result is useful for estimating Poisson probabilities when the mean is large. It also explains why the normal approximation has mean λ and variance λ, and why a continuity correction is used when approximating probabilities for integer counts.

The convergence statement

Let $X_\lambda \sim \operatorname{Poisson}(\lambda)$. Its mean and variance are both λ, so its standard deviation is $\sqrt{\lambda}$. Then, as $\lambda \to \infty$,

The arrow $\xrightarrow{d}$ means convergence in distribution. Equivalently, for every real number $x$,

where $\Phi$ is the cumulative distribution function of the standard normal distribution.

In unstandardized terms, this says that for large λ, $X_\lambda$ is approximately normal with mean λ and variance λ:

The parameter λ need not be an integer. The limit holds as the Poisson mean increases through positive real values.

Why the normal limit appears

The moment-generating function of a Poisson variable is $M_X(s)=\exp\{\lambda(e^s-1)\}$. For the standardized variable $Z_\lambda=(X_\lambda-\lambda)/\sqrt{\lambda}$, its moment-generating function is

To see the limit, expand $e^{t/\sqrt{\lambda}}$: after the linear term cancels, the exponent begins with $t^2/2$, while the next term is $t^3/(6\sqrt{\lambda})$ and vanishes as λ grows. The limiting function $\exp(t^2/2)$ is the moment-generating function of $N(0,1)$, which gives the stated convergence.

There is also an intuitive connection to the central limit theorem. When λ is an integer, a Poisson$(\lambda)$ variable can be written as the sum of λ independent Poisson$(1)$ variables. A sum of many independent contributions, when centered and scaled, is the setting of the central limit theorem. The moment-generating-function argument also covers noninteger λ.

Approximating a Poisson probability

For a large mean, approximate $X_\lambda$ by a normal variable with the same mean and variance. Since the Poisson variable is discrete and the normal variable is continuous, shift integer cutoffs by half a unit. For example,

For an interval $a\le X_\lambda\le b$, approximate it using the normal interval from $a-0.5$ to $b+0.5$.

Worked example: a Poisson mean of 100

Suppose $X\sim\operatorname{Poisson}(100)$ and you want to estimate $\Pr(X\le110)$. The normal approximation is $N(100,100)$, with standard deviation 10. Apply the continuity correction and standardize:

So the normal approximation estimates the probability at about 0.853. Without the continuity correction, the calculation would use $z=(110-100)/10=1$ and give about 0.8413. The correction matters because it accounts for the full width of the integer count at the boundary.

When the approximation is useful

The approximation improves as λ increases, because the Poisson distribution becomes less skewed. Its skewness is $1/\sqrt{\lambda}$, which tends to zero. There is no single cutoff that guarantees adequate accuracy for every problem: a rough approximation may be acceptable at a smaller mean, but a tail probability or a high-precision answer may require a larger mean or an exact Poisson calculation.

The approximation is often reasonable for probabilities near the center of the distribution once the mean is moderately large. Be more cautious in extreme tails, where small differences in the shape can have a noticeable effect on the probability being estimated.

Common mistakes

• Leaving the variable unstandardized in the limit statement. The variable that converges to $N(0,1)$ is $(X_\lambda-\lambda)/\sqrt{\lambda}$, not $X_\lambda$ itself. The unstandardized approximation is $N(\lambda,\lambda)$, whose mean and variance change with λ.
• Using $\sqrt{\lambda}$ as the variance. It is the standard deviation; the variance is λ.
• Forgetting the continuity correction. A Poisson probability is a sum over integer values, so half-unit adjustments generally give a better normal approximation to count probabilities.
• Treating the approximation as exact. For a small mean or a probability far into a tail, calculate the Poisson probability directly when possible.
• Assuming a fixed Poisson distribution converges just because sample size grows. The normal limit here requires the Poisson mean itself to grow. A fixed $\lambda$ describes one fixed distribution.