General Report on Tunny

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21 Page 42

success. Then the probability of exactly r successes is This is the so-called binomial distribution.

	If

we say that x has a normal (or Gaussian) distribution. It is easy to see that the mean of x is x and its S.D. is σ. The factor is the so called normalising factor which makes .

The integral of ƒ(x) is called the error function. A convenient way of tabulating this is in a deciban form. A table of Ψ(x) is given in R1, 109 where

The binomial distribution is closely approximated by the normal one for quite small values of n, if we take x = pn and . In Tunny theory this is the most frequent form for σ. The normal distribution is also a good approximation when a variable is the sum of a lot of small independent contributions.

If the probability of exactly n successes is e^-aaⁿ/n :, n is said to have a 'Poisson Distribution'. The formula is easy to remember since aⁿ/n! is a typical term of the expansion of aⁿ, so that The average and variance of the distribution are both equal to a.

The binomial distribution is approximated by the Poisson distribution if n is fairly large but p is small, so that the average is much less than n. The Poisson distribution is approximated by the normal distribution when the number of successes minus a is small compared with a.

There is one other distribution used in the research logs, namely the 'Χ² distribution'. Given n independent variables each with a normal distribution of mean 0 and S.D. 1, let Χ² be the sum of the squares of these variables.

writing	we have
where

This is the most convenient formula when t is a good deal larger than n, as it is in all our applications.

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