Correct adjective for an RV with a Student's $t$-distribution If a variable has a normal distribution, we say it is normally distributed. If it has an exponential distribution, we say it is exponentially distributed.
What is the proper way to reference a variable with a Student's $t$-distribution?
For example, how would one complete this sentence:
"... and we consider the observations to be <> random variables"?
 A: "Normal" &  "exponential" are adjectives, already furnished with the adverbs "normally" & "exponentially". When a noun or noun phrase is pressed into service as an adjective there won't in general be a corresponding adverb to hand†. So "Weibull distributed" or "gamma distributed", though you sometimes see them, are already doing some violence to grammar (I'd avoid those). "Student's t distributed" seems even worse; I think because the possessive modifier is supposed to apply to the "t distribution", or if not just because it reminds us that "Student's t" is really a noun phrase&ddagger;. So  don't say that.
"Student's t random variable" seems fine—here "t" is being used attributively, as an adjective; & why not? "Has a Student's t distribution" or "follows a Student's t distribution" are other options.
† Even when we've a proper adjective, like "Gaussian", we're still not guaranteed an adverb. "Gaussianly distributed"? No.
&ddagger; I'm uncertain whether "Student's t distribution" is better parsed as "the t distribution, invented by Student", or "the distribution of Student's t statistic".
A: Scortchi's prescriptive answer is very good. From a purely descriptive viewpoint, I have found all of the following on Google scholar, but my conclusion is these forms are relatively unusual, particularly when we compare how often a phrasing with "normally distributed" is used instead of "normal distribution" — in comparison to the relatively small numbers found below, there were tens of thousands of results for both "Student's t distribution" and "Student t distribution".


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*"Student's t distributed" found 154 papers, including the particular forms (I include a brief snippet for context):


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*generate a Student's t distributed signal (no hyphen; this variant seems most common)

*modelled as an EGARCH(1,2) process with Student's t-distributed
innovations (hyphen between "t" and "distributed"; still fairly common)

*the innovations ... are fat- tailed as well, and are Student's-t distributed (hyphen between "Student's" and "t"; this variant noticeably less common than the other two)


*"Student t distributed" found just over 1,000 papers, though this included many results with "Student's".


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*a GARCH model under the assumption of Student-t distributed errors (hyphen between "Student" and "t"; this variant most common by a clear margin)

*the loss L is Student t distributed (no hyphen)

*independent Student t-distributed random variables (hyphen between "t" and "distributed"; this variant fairly rare)

*with Student-t-distributed innovation (hyphens between "Student" and "t" and "distributed"; this variant least common by a clear margin)


*"Student distributed" had about 500 results, of which many were false hits (e.g. "a student distributed the surveys"). So certainly less common than variants of "Student t distributed" but probably at least as common, if not more so, than "Student's t distributed".


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*This produces returns that are Student distributed for intraday time scales (no hyphen, by far the most common)

*The Student-distributed random variables were obtained (with hyphen, unusual, but note there may be grammatical reasons for placing a hyphen in a compound modifier when it appears before the noun phrase it is describing, as in the cited example.)


*"Student's distributed" had just ten results, including false hits for a similar reason as "student distributed" did. There were results both with and without hyphens but the sample size is too small to determine which is more common.


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*the fractional power change rate (or gradient) is Student's distributed with two degrees of freedom (without hyphen)

*the Student's-distributed factor analyzers (with hyphen)


*I even turned up 60 papers with "t-Student distributed", for instance "Simulated sizes of the test for t-Student
distributed errors", where there was invariably a hyphen between "t" and "Student". This looks odd to me, so I wouldn't recommend it, but it was not particularly rare. I did check and found a fair number of papers using the phrase "t-Student distribution" so perhaps reversing "t" and "Student" is not as unusual as it strikes me (Google Scholar found 2000 hits, compared to 11,000 for "Student's t distribution" and 25,000 for "Student t distribution").


So actual practice varies widely! I think the most common variant I found overall was "Student-t distributed", and if you were to go with this then your usage would not be unusual. Something I found puzzling was how the location of the hyphen is very inconsistent: while "Student's t-distributed" is more common than "Student's-t distributed" (which does look very unnatural), when the "'s" is omitted I found that "Student-t distributed" was massively more common than "Student t-distributed". It seems that people feel a hyphen is needed somewhere, but do not necessarily agree on where it should go. On the other hand, before "distributed" the "Student t" or "Student's t" is acting as an adverb, and style guides generally declare along the lines that "most adverbial phrases do not need hyphens".
