Moment calculations
Having the joint density, we can calculate moments such as
for some positive integers l and m.
Using the density expression with r = 2,
Changing variables to u = x1 and v = (1 - x1 - x2)∕x2,
over the region u > (1 - u)∕(1 + v) ≥ 0 and v ≥ 0.
Let w = max(0, (1 — 2u)∕u). Intherangeof u > 1/2, w = 0. Then, a lower bound on J is obtained as
where H is a constant defined by
See Watterson and Guess (1977) for approximate evaluations of these moment expressions.
10.8