Giving an example of a t-distribution with independent $X_1, X_2, X_3$

Question

Let Xi be a random variable distributed as $N(i, i^2), i = 1, 2, 3$. As-sume that the random variables $X_1, X_2$, and $X_3$ are independent. Using only the three random variables $X_1, X_2$, and $X_3$ give an example of a statistic that has a t distribution with two degrees of freedom.

The definition of a t-distribution that we use in my class is $T= \frac{Z}{\sqrt{U/r}}$ where $U$ is $\chi^2(r)$.

I came up with $\frac{X_1/i}{\sqrt{(X_1+X_2)/2}}$ as an answer, but I just wanted to verify that it's valid or if I'm doing the wrong thing entirely.

Answer 1

I doubt that your proposed $T \sim \mathsf{T}(2)$.

First, numerator and denominator are not independent.

Second, in denominator $(X_1 + X_2)$ is not chi-squared; worse it can take zero and negative values. So even if you achieved independence by using something like$(X_2 + X_3),$ instead of $(X_1 + X_2),$ the idea still wouldn't work.

See my Comment for something that does work.