AP Statistics Curriculum 2007 Limits Norm2Poisson

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Contents 1 General Advance-Placement (AP) Statistics Curriculum - Normal Approximation to Poisson Distribution 1.1 Normal Approximation to Poisson Distribution 1.2 Examples 1.3 References

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

Normal Approximation to Poisson Distribution

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

For sufficiently large values of λ, (say λ>1,000), the Normal(μ = λ,σ² = λ) 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.

If x_o is a non-negative integer, and ), 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 λ = 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: 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 , together with the continuity correction for better approximation we obtain , which is close to the exact that was found earlier. The figure below shows this probability.

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References

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

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