Suffix in adverbs: The use of the term 'normally distributed' in place of 'normal distributed'

Question

When a word is used as an adverb then it is normally combined with a suffix like "-ly" or "-ian".

Like 'normal' becomes 'normally' and the previous sentence is an example of its use.

In statistics, common phrase have a construction like 'X is normally distributed', 'X is binomially distributed' or 'X is geometrically distributed'.

But less common are phrases like 'X is Poissonian distributed' or 'X is Bernoullian distributed'. (exception 'X is Gaussian distributed').

• Why does the normal/Gaussian distribution get the use of an adverb-related suffix while this is not done as much with other distributions, especially with distributions that are named after a person?

• Is the use of a suffix to make the root word an adverb correct?

  The meaning 'X is normally distributed' is supposed to refer to 'X is distributed as a normal distribution' or 'X follows a normal distribution' (the normal distribution is the name of a particular type of probability distribution, it does not refer to 'normal' as in 'usual' or 'common'). There are also phrases like 'evenly distributed', but that relates more to the way that the distributing is done.

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Some statistics about the use on the website stats.stackexchange.com is found with this query which allows counting the occurrences of sentence constructions like "is ... distributed".

With distributions that are named after a person the use of a suffix is less common.

	with suffix	without suffix
normally/normal	610	19
geometrical/geometric	4	1
binomially/binomial	22	5

	with suffix	without suffix
Gaussian/Gauss	10	0
Poissonian/Poisson	0	33
Bernoullian/Bernoulli	0	9

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Hits on google.scholar

	with suffix	without suffix
Gaussian/Gauss	16 600	169
Poissonian/Poisson	92	15 900

Answer 1

The constructions "Gaussian distributed", "Poisson distributed", "Bernoulli distributed" are formed by analogy with the more frequent constructions ending in the noun "distribution": "Gaussian distribution", "Poisson distribution", Bernoulli distribution". "Gaussian distributed" is like a short form of saying "distributed according to a Gaussian distribution".

"Poisson distribution" and "Bernoulli distribution" follow the common noun-phrase-forming pattern noun + noun. They could be interpreted as compounds, or as phrases starting with an "attributive noun". "Gaussian distribution" follows the common noun-phrase-forming pattern adjective + noun. The adjective "Gaussian" is a proper adjective formed from the proper name "Gauss". When to use an attributive proper noun vs. a proper adjective is not a straightforward topic; often it comes down to convention with no easy way to explain general principles.

The rule seems to be that if the of the form used before the noun “distribution” is a non-proper adjective, it is often turned into an adverb in the corresponding expression ending in “distributed”. But if the form used before “distribution” is a proper noun or proper adjective, it remains as it is. Likely, this is for the reason Sextus Empiricus identifies: the adverb-forming suffix -ly does not normally appear on nouns, and often sounds awkward on proper adjectives. But "normally" does not sound awkward because "normal" is an ordinary adjective that already had a commonly used adverb form before the expression "normal distribution" came into use.

The Google Ngram Viewer actually shows a small amount of usage of "Gaussianly distributed", with an adverb formed regularly from the adjective "Gaussian" + the suffix "-ly".