Final answer:
The variance of the t-distribution with v degrees of freedom for v > 2 is calculated to be v/(v - 2) by making a substitution in the defining integral, resulting in a simplified expression evaluatable using calculus.
Step-by-step explanation:
To show that for v > 2 the variance of the t-distribution with v degrees of freedom is v/(v - 2), we can start by recognizing that the variance is defined for the t-distribution only when the degrees of freedom are greater than two. The variance of the t-distribution (which is a standardized form of the Student's t-distribution) with v degrees of freedom is calculated by a specific integral that takes into account the shape of the t-distribution curve.
By substituting 1 + t²/v = u into the integral that defines the variance of the t-distribution, the integral simplifies and allows us to solve for the variance. This proof involves calculus and knowledge of probability distributions. After performing the substitution and evaluating the integral, one arrives at the result that the variance of the t-distribution with v degrees of freedom is indeed v/(v - 2) for v > 2.