Let $Y\sim ST(\mu,\sigma^2,\lambda,\nu)$ where ST means Skew-t distribution. Find $\operatorname{E}(Y)$ and $\operatorname{Var}(Y)$.
I'm litle lost in how to do it. What I know is below
Let $X\sim SN(0,\sigma,\lambda)$ be a skew-normal random variable and $U\sim \operatorname{Gamma}(\nu/2,\nu/2)$ then $$Y=\mu+U^{-1/2}X\sim ST(\mu,\sigma^2,\lambda,\nu)$$
but it don't make things more easier. Then I have the following stochastic representation of ST $$Y\mid T=t,U=u\sim N(\mu+\Delta t,\mu^{-1}\Gamma)$$ $$T\mid U=u\sim HN(0,\mu^{-1})\qquad\text{Half-normal}$$ $$U\sim \operatorname{Gamma}(\nu/2,\nu/2)$$ where $\Delta=\sigma\delta$, $\Gamma=\sigma^2(1-\delta^2)$ and $\delta=\frac{\lambda}{\sqrt{1+\lambda^2}}$
The results say that
$$\operatorname{E}[Y]=\mu + \left( \frac \nu \pi \right)^{1/2} \frac{\Gamma((\nu-1)/2)}{\Gamma(\nu/2)} \sigma\delta\qquad \nu>1$$
$$\operatorname{Var}(Y)=\frac{\nu}{\nu-2}\sigma^2-\frac{\nu}{\pi} \left(\frac{\Gamma((\nu-1)/2)}{\Gamma(\nu/2)} \right)^2 \sigma^2\delta^2$$
Anyone can give me any tip?