# 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
Commented Jan 11, 2017 at 23:04
• @Michael The Cauchy distribution is the Student t with 1 df.
– whuber
Commented Jan 12, 2017 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 $-\infty$ Commented Jan 12, 2017 at 0:41
• Commented Aug 14, 2017 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. Commented Aug 15, 2017 at 10:20