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The variance of a t-distribution is given by df/(df-2), hence the t-distribution with 1 and 2 degrees of freedom have no defined variance. Yet these distributions do exist and can be drawn, so one would say that their variances can be calculated. I hope somebody can explain this without using too much mathematics, as my skills in mathematical statistics are poor.

user145023
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  • Have you seen Douglas Zare's nice (physical) explanation at http://stats.stackexchange.com/questions/36027/why-does-the-cauchy-distribution-have-no-mean/36037#36037? The way you ask your question also suggests you might be conflating two meanings of "variance": one is the variance of a hypothetical data-generation distribution while the other is the variance of a dataset. The former can be infinite while (obviously) the latter is always finite. – whuber Jan 11 '17 at 23:04
  • The t with 1 and 2 degrees of freedom are legitimate probability distributions. They have very heavy tails which is why the variances don't exists. The Cauchy distribution is also a heavy tailed symmetric distribution and it doesn't even have a first moment. – Michael R. Chernick Jan 11 '17 at 23:26
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    @Michael The Cauchy distribution is the Student t with 1 df. – whuber Jan 12 '17 at 00:01
  • Thanks I didn't know that. So that means t with 1 degree of freedom doesn't have a mean. I think of the Cauchy as the ratio of two independent standard normals. – Michael R. Chernick Jan 12 '17 at 00:14
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    When you "draw the distribution" are you drawing all of it? Or only the middle 99.something percent of it? It's not that part that makes the variance not-finite. The finiteness or otherwise of the variance is essentially to do with the way the tail behaves in the limit as the variable approaches $\infty$ and $-\infty$ – Glen_b Jan 12 '17 at 00:41
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    Related: https://stats.stackexchange.com/questions/94402/what-is-the-difference-between-finite-and-infinite-variance/100161#100161 – kjetil b halvorsen Aug 14 '17 at 21:00
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    I'm not convinced this is an exact duplicate because I think it would be helpful to see an answer that covers the case with two degrees of freedom. Clearly one degree of freedom is already covered. – Silverfish Aug 15 '17 at 10:20
  • @Silver Where, then, do we stop? When somebody asks about $1.5$ df, should we allow that to stand as a unique question, too? – whuber Aug 18 '17 at 16:02
  • @whuber People usually deal (I know, not exclusively, but usually) with integer degrees of freedom. Of those, $t_1$ and $t_2$ are the only two special cases in this regard. In my opinion, if there were a thread for the most common special cases, and an answer there or elsewhere that is already generalized to deal with other cases (not merely "easily generalized" but preferably already and explicitly generalized, since "easily" is very much in the eye of the beholder) then I'd have no hesitation closing "1.5 df" as a duplicate. But 2 df perhaps deserves its own little place under the sun – Silverfish Aug 18 '17 at 17:12
  • If the complaint is that 1 df and 2 df do not deserve separate threads (though plenty of people searching for "Cauchy" will not be thinking of it as a t with 1 df), then an alternative would be answering both cases at this thread, and merging the threads with the other one left as a duplicate. – Silverfish Aug 18 '17 at 17:15
  • @Silver One (IMHO) fairly thorough answer to the general question of determining which moments exist for any distribution has appeared at https://stats.stackexchange.com/questions/289321/how-to-prove-whether-the-mean-of-a-probability-density-function-exists/289347#289347. As far as "usually" goes, conventional wisdom (and the R default!) is to use a Welch test to compare two means, which almost always uses a Student t distribution with nonintegral df. Thus, your sense of "usually" might not be quite correct. – whuber Aug 18 '17 at 17:54
  • @whuber That's a fair comment - I had Welch in mind with "not exclusively". But people taking an introductory course in mathematical statistics (a likely target audience for this question) will usually be dealing with integer degrees of freedom when they come across the $t$-distribution, and it is the well-known and important cases of $t_1$ and $t_2$ which stick out like sore thumbs when you are being taught its properties for the first time. I suspect most people first coming across the Welch test do so on applied statistics courses & likely have less interest in the mathematical properties. – Silverfish Aug 18 '17 at 19:09

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