Angels on pinheads

I was going through an A level exam paper with a student this morning and we looked at a question which boiled down to:

Using a suitable approximation, find P(X≥10) given that X~B(108,2/27).

The mark scheme indicated that the Normal approximation to the Binomial was expected, but I suspected the Poisson approximation to the Binomial would be better. As it turns out, I was right. The exact probability, using the Binomial distribution we were given, is 0.2784 to 4 decimal places. The Poisson approximation gives 0.2833 and the Normal approximation gives 0.2908.

Why was the Normal distribution less good? After all, the mean (np) and variance (npq) of the Normal distribution I used were the same as the mean (np) and variance (npq) of the Binomial distribution I was approximating. Whereas the variance of the Poisson distribution (np) was larger than the variance of the Binomial distribution (npq), which it always will be when using the Poisson approximation because q<1.

Well, the mean and the variance are not the only descriptors of the shape of a distribution: we can also look at the skew and the kurtosis. The Normal distribution is symmetric and therefore has zero skew. The Binomial distribution we are approximating is slightly skewed to the right, as are all Binomial distributions with p<1/2. The Poisson distribution is also always skewed to the right, and this suggests it may be a better fit than the Normal.

Similarly, the Normal distribution has kurtosis equal to 3, but both the Binomial (in this case) and the Poisson (always) have kurtosis greater than 3, again indicating that the Poisson may be the better fit.

You can see how the three distributions compare in the diagram above. (You can click on it to bring up a larger version.) Although the Binomial and Poisson distributions are discrete, I have drawn them as if they were continuous. The Binomial distribution is given in purple, the Poisson distribution in red and the Normal distribution in blue. As you can see, the Binomial and Poisson appear to lean over to the left slightly compared with the Normal distribution. It is less easy to see the effect of kurtosis.

What does seem to be clear is that the Poisson distribution is a far better fit in this case.

As for the title of this blog entry: well, does it really matter? I'd make two points. First, although distribution approximations are interesting and help with understanding of the derivation of the Poisson distribution and the Central Limit Theorem, we don't really ever need to use them. The probabilities I calculated above were found using a fairly ordinary calculator that has been on the market for nearly two decades.

Second, why do we quote probabilities to four decimal places? Why do we need them that accurately? And could we really tell the difference between a probability of 0.2784 and one of 0.2908? We would likely need many thousands of observations until the difference became meaningfully apparent.

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