Homework Solutions #5
This density is symmetric about zero, hence the mean is zero. Hence there
is no difference between central moments and ordinary moments and
.
Now
This density is symmetric about zero, hence the mean is zero and . Now
This density is symmetric about 1/2, hence the mean is 1/2.
Also
Since
,
we also have
,
but
Almost exactly the same calculation as the preceeding problem, except
one starts with the inequality
This was CANCELLED, because it turned out to be messier than I thought.
We can get rid of one sum. There are n terms with i = j hence ,
and there are 2 (n - 1) terms with
hence ,
and there are 2 (n - 2) terms with
hence ,
and so forth to
2 terms with i = 1 and j = n or vice versa hence
,
thus
but this does not simplify any further, at least not using the geometric
series.
If anyone is wondering how I ever thought this was simple, I was recalling that the limit as n goes to infinity is simple
Using the linear combination form for variance, we have
because
First we need to do the analogous equation for covariance,
which isn't given in the notes or in Lindgren.
Then
We want to use
Note the density is
Then
and this cannot be simplified if p is not an integer.
Using part (a) and the recursion formula for the gamma function, (B.2)
in the appendix on ``brand name distributions'' of the notes, twice
The integral
The question asked about positive integers, so the answer is k = 1 or 2.
For k < 3
Note (not a part of the problem, but an interesting point) that the formula
The integral
Thus E(Xk) exists for k = 1, 2, (all positive integers).