# why is the variance of t-distribution with 1 and 2 degrees of freedom undefined while these distributions can be drawn? [duplicate]

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.

• Have you seen Douglas Zare's nice (physical) explanation at stats.stackexchange.com/questions/36027/…? 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
• @Michael The Cauchy distribution is the Student t with 1 df. – whuber Jan 12 '17 at 0:01
• 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 0:41
• – kjetil b halvorsen Aug 14 '17 at 21:00
• 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