AP Statistics Curriculum 2007 Limits Norm2Poisson

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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. Suppose P(Xx), 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 PX(X < xo) = PU(U < xo + 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:  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




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