Variance of a t-distribution

Problem

The image below displays the pdf a t-distribution with 1, 2, 5 and \(∞\) degrees of freedom. The t-distribution with \(∞\) dof is the same as the standard normal.

1. Based on the image, which one has the highest variance: the standard normal or the t-distribution with 5 degrees of freedom?
2. Confirm your answer to a) by finding the variance of the t-distribution. You can find it using Wolfram alpha, “variance of student t 5 dof”.

(If you try to do the same for 1 or 2 dof you get the answer “undefined”. The tails of the t- distribution with 1 and 2 dof are so fat that integral used to calculate the variance becomes infinite. The t- distribution with 1 and 2 dof simply have no variance. The t-dist with 1 dof does not even have an expected value).