# AP Statistics Curriculum 2007 Limits Norm2Poisson

## Contents

## General Advance-Placement (AP) Statistics Curriculum - Normal Approximation to Poisson Distribution

### Normal Approximation to Poisson Distribution

The Poisson(\( \lambda \)) distribution can be approximated with Normal when \( \lambda \) is large.

For sufficiently large values of λ, (say λ>1,000), the Normal(\(\mu=\lambda, \sigma^2=\lambda\)) distribution is an excellent approximation to the Poisson(λ) distribution. If λ is greater than about 10, then the normal distribution is a good approximation if an appropriate continuity correction is performed. Suppose P(*X*≤*x*), where (lower-case) *x* is a non-negative integer, is replaced by P(*X* ≤ *x* + 0.5).

- \[F_\mathrm{Poisson}(x;\lambda) \approx F_\mathrm{normal}(x;\mu=\lambda,\sigma^2=\lambda)\,\]

If \(X\sim Poisson(\lambda)\) and \(U\sim Normal(<math>\mu=\lambda, \sigma^2=\lambda\)), then \(P_X(X<x_o) = P_U(U<x_o+0.5)\).

### Examples

Suppose cars arrive at a parking lot at a rate of 50 per hour. Let’s assume that the process is a Poisson random variable with \( \lambda=50 \). Compute the probability that in the next hour the number of cars that arrive at this parking lot will be between 54 and 62. We can compute this as follows\[ P(54 \le X \le 62) = \sum_{x=54}^{62} \frac{50^x e^{-50}}{x!}=0.2617. \] The figure below from SOCR shows this probability.

**Note**: We observe that this distribution is bell-shaped. We can use the normal distribution to approximate this probability. Using \( N(\mu=50, \sigma=\sqrt{50}=7.071) \), together with the continuity correction for better approximation we obtain \( P(54 \le X \le 62)=0.2718 \), which is close to the exact that was found earlier. The figure below shows this probability.

### References

- SOCR Home page: http://www.socr.ucla.edu

Translate this page: