Let $(X_n)_n$ a suite of random variables independent and identically distributed, $X_i \sim \mathcal{N}(0,1)$ and let $Y_n:= \sum_{j=1}^n X_j ^2 \sim \chi^2_n$ a chi-square random variable with $n$ degree of freedom, and let \begin{align} T_n := \sqrt{n} \frac{X_{n+1}}{\sqrt{Y_n}} \end{align} I showed that $T_n$ is a Student random variable of parameter $n$. I have to calculate $\mathbb{E}(T_n)$ and $Var(T_n)$ without using the probability density function of $T_n$.
For the expected value I have that \begin{align} &\mathbb{E}(X_{n+1}) = 0 \, , \quad \mathbb{E}(Y_n) = n \\ &\mathbb{E}(T_n) = \sqrt{n} \cdot \mathbb{E} \left( \frac{X}{\sqrt{Y_n}} \right) \end{align} but I have some problem to compute the expected value of the ratio $\frac{X}{\sqrt{Y_n}}$.
Any suggestions? Thanks in advance!