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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!

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  • $\begingroup$ Your definition of t-distribution is not correct. $\endgroup$ – StubbornAtom May 1 at 20:38
  • $\begingroup$ Sorry, $X \sim \mathcal{N}(0,1)$ $\endgroup$ – userr777 May 1 at 20:39
  • $\begingroup$ Still incomplete, unless you add independence. That allows you to calculate the expected value and variance you are after. $\endgroup$ – StubbornAtom May 1 at 20:40
  • $\begingroup$ @StubbornAtom Sorry again, hope it's correct now $\endgroup$ – userr777 May 1 at 20:46
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    $\begingroup$ 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. $\endgroup$ – Henry May 1 at 21:06

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