Normal distribution hazard function continued fraction

The hazard function of a probability distribution is the instantaneous probability density of an event given that it hasn’t happened yet. This works out to be the ratio of the PDF (probability density function) to the CCDF (complementary cumulative density function).

For the standard normal distribution, the hazard function is

$h(x) = \frac{\exp(-x^2/2)}{\int_x^\infty \exp(-t^2/2)\,dt}$

and has a surprisingly simple continued fraction representation:

$h(x) = x + \cfrac{1}{x+\cfrac{2}{x+\cfrac{3}{x+\cfrac{4}{x+\cdots}}}}$

Aside from being an elegant curiosity, this gives an efficient way to compute the hazard function for large x. (It’s valid for any positive x, but most efficient for large x.)

Source: A&S equation 26.2.14

Typesetting and computing continued fractions
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Normal hazard continued fraction

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