MSV 30: A Close Approximation

www.making-statistics-vital.co.uk MSV 30: A Close Approximation

Homer is using a Binomial distribution to model a random variable X. He says X ~ B(n, 0.01), where n is large. He approximates this with the Poisson distribution Po(n × 0.01).

Homer finds that P(X = 2) calculated using B(n, 0.01) and P(X = 2) calculated using P(n × 0.01) are extremely close. In fact, for no value of n could they be closer. What is n in this case?

Answer If X ~ B(n, 0.01), then P(X = 2) = (0.01)2(0.99)n-2n(n-1)/2. If X ~ P(n × 0.01), then P(X = 2) = e-0.01n(0.01n)2/2. Plotting y = (0.01)2(0.99)x-2x(x-1)/2 - e-0.01x(0.01x)2/2 gives this:

There are two possible points at which the curve and the x-axis cross. Zooming in shows that these points are close to x = 59 and x = 341. The value of y at 59 is 3.51 × 10-6, and y at 341 is -1.37× 10-7. So since ‘for no other value of n could they be closer,’ X ~ B(341, 0.01) ≈ P(3.41). For P(X=2), the approximation is 2.63 × 10-6 % out.

For X ~ B(341, 0.01), P(X = 1) = 0.11187… For X ~ P(3.41), P(X = 1) = 0.11267... So approximation is 0.71% out here.

With thanks to the Simpson family www.making-statistics-vital.co.uk is written by Jonny Griffiths hello@jonny-griffiths.net

MSV 30: A Close Approximation