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,σ^{2} = 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(X≤x), where (lower-case) x is a non-negative integer, is replaced by P(X ≤ x + 0.5).
If 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.
References
- SOCR Home page: http://www.socr.ucla.edu
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