Need clarification from an already answered question.

Question

I was reading this question (Proving $E[X^4]=3σ^4$). It would be really helpfull to know why $X=\sigma U$ (Please, see the comments in the question). Where does $\sigma$ come from? I have no problems on why $E(U^4)=3$, tho.

Answer 1

Define $U$ as $\frac 1 {\sigma } X$. Then $var (U)=EU^{2}-(EU)^{2}=\frac 1 {\sigma^{2} } EX^{2}-(EX)^{2}=\frac 1 {\sigma^{2} } var (X)=1$. Hence $U$ has $N(0,1)$ distribution and we can calculate $EU^{4}$ by the method described in your link.