Prove that this has t-distributionnwith k degrees

Definition 2.5. A continuous r.v. X is said to have a Students t-distribution with k degrees of freedom (d.f.) if, and only if, the PDF of X is given by k+1 2 m m/2 X2 fx(2) = 1 + 1(-00,00) (2) , KE Z+ (VIT ) I k k Notation: X ~ t(k) Mean: E(X) = 0 k Variance: Var(X) = k > 2 k - 2' MGF: DNE 23