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MSV 30: A Close ApproximationPowerPoint Presentation

MSV 30: A Close Approximation

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MSV 30: A Close Approximation

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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