It is known that the expected value of the F-Distribution depends only on the degrees of freedom associated with the Chi-square random variable in the denominator.
In other words, E[1/X] does not equal 1/E[X].
Could someone help me intuitively understand why this is?
This is a consequence of Jensen's inequality, as $1/x$ is a convex function. Consider, to gain some intuition, a distribution giving probability mass to only two points, $a$ and $b$. Let $g(x)$ be a convex function. Then,
$$ g(E[x]) = g(p(a)g(a) + p(b)g(b))$$
the value of $g(x)$ at some point $x$ between $a$ and $b$. However,
$$E[g(x)] = p(a)g(a) + p(b)g(b) $$
is a value in the segment from $(a,g(a))$ to $(b,g(b))$ and said segment is above $g(x)$ (by the convexity of $g(x)$): so $E[g(x)] \ge g(E[x])$.
