# Calculate expected value and variance of a t-student distribution without using density function

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 $$\frac{X}{\sqrt{Y_n}}$$.

• Sorry, $X \sim \mathcal{N}(0,1)$ – userr777 May 1 at 20:39
• There is a symmetry argument on $X_i$ that means $T_n$ also has a symmetric distribution about $0$ and so either has a mean of $0$ or does not have a mean. – Henry May 1 at 21:06