February 2015 Archives

Characteristically Pathological

In the last post we took a look at the log normal distribution, the statistical distribution that describes the outcomes of products of random variables.
Whilst the PDF, CDF, inverse CDF and random number generator were trivially implemented by combining logarithms and exponentials with the normal distribution's equivalent functions, it proved to be rather more difficult to implement the characteristic function...

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The Product Distribution Business

Several posts ago we covered the normal distribution; the statistical distribution that governs the behaviour of sums of random variables by the central limit theorem. We noted that by the properties of logarithms we could transform a product into a sum with

  ln(x0 × x1 × x2 × ...) = ln x0 + ln x1 + ln x2 + ...

and consequently the distribution that governs the behaviour of products of observations of a random variables is the normal distribution of their logarithms.
Since many sources of randomness accumulate from the multiplication, rather than the addition, of random components, this is an incredibly useful observation. So much so that it is an important distribution in its own right; the log normal distribution, LN(μ,σ).

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