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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.

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  • $\begingroup$ My rote response would have been to standardize $Z_i = (X_i - i)/i$ then to say $T = \frac{\bar Z - 0}{S_Z/\sqrt{3}} \sim \mathsf{T}(\nu = 2).$ $\endgroup$ – BruceET Feb 18 '17 at 23:24
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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.

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