We can create an F-distribution if we have two independent chi-squared random variables, and we divide them by their own degress of freedom, and divide the result by each other.

\(F = \frac{\chi^{2}_{1}/\nu_{1}}{\chi^{2}_{2}/\nu_{2}}\).

We have that:

\(E(\chi^{2}_{1}/\nu_{1})=1\)

\(E(\chi^{2}_{2}/\nu_{2})=1\)

Now intuitively I would then think that E(F)=1, but we have

E(F) = \(\frac{\nu_{2}}{\nu_{2}-2}\)

This can offcourse be proven mathematically, but is there an intuitive way of seeing that the expected value can not be 1, even though each part of the fraction has expected value 1? And is it an intuitive way to see that it has to be greater than 1, not \(\le 1\)?