Statistics : Where did this degree of freedom formula for the T distribution come from?

I am on the hypothesis testing for two populations unit. I need some intuitive explanation as to why this formula is used. My statistics professor put this up on the board but he didn't explain why its true.

For a T distribution, the formula for the degrees of freedom is:

$$ \large \mathrm{df} = \frac{ \left(\frac{s_{1}^{2}}{n_1} + \frac{s_{2}^{2}}{n_2} \right)^{2} }{ \frac{\left(\frac{s_{1}^{2}}{n_1}\right)^2}{n_{1} - 1} + \frac{\left(\frac{s_{2}^{2}}{n_2}\right)^2}{n_{2}-1}}$$

Here $s_1, s_2$ are the sample standard deviations and $n_1,n_2$ are the sample sizes.

Solution 1:

This question is essentially answered. The comment of user "cardinal" that mentions the B.L. Welch (1947) paper provides all there is to it, as regards where the formula comes from. Welch derives the exact mathematical solution for the general problem of calculating the degrees of freedom when the samples are more than two, then he develops an approximate solution through a Taylor expansion, and then mentions the resulting approximate df-formula for the special case of two samples. There is no deep intuition behind the formula, just patient (but healthy) mathematics. Welch's style and notation is rather old-fashioned - for educational purposes, another paper of his "The Significance of the Difference Between Two Means when the Population Variances are Unequal", Biometrika, Vol. 29, No. 3/4 (Feb., 1938), pp. 350-362, focuses on the two-samples case, and is a bit more accessible.