> I continue to have problems solving for zeroes of the
> derivatives, apparently because the summation is not being
> broken up into multiple summations.
Yes, that is the problem. Try declare(sum,linear), then resimplifying
using expand, and *then* solving. See transcript below.
By the way, it is best to remove tabs from Maxima displays in email,
since different systems display tabs differently, and then when your
text is quoted with ">", it gets even more garbled. Alternatively, show
the linear form in Maxima (using display2d:false).
--------------------------------------------
(C1) declare(sum,linear);
(D1) DONE
(C2) loglik_1 : 'log(1/sqrt(2 * %pi) * 1 / sigma * exp(-1/(2*sigma^2) *
(x[i] - mu)^2));
2
(x - MU)
i
- ----------
2
2 SIGMA
%E
(D2) LOG(-----------------------)
SQRT(2) SQRT(%PI) SIGMA
(C3) loglik : 'sum(loglik_1, i, 1, n);
2
(x - MU)
i
- ----------
n 2
==== 2 SIGMA
\ %E
(D3) > LOG(-----------------------)
/ SQRT(2) SQRT(%PI) SIGMA
====
i = 1
(C4) diff(loglik,mu);
n
====
\
> x - MU n
/ i
====
i = 1
(D4) ---------------
2
SIGMA
(C5) solve(%,mu);
n
====
\
> x
/ i
====
i = 1
(D5) [MU = --------]
n
(C6) diff(loglik,sigma);
2
(x - MU)
i
- ----------
2
n 2 2 SIGMA
==== (x - MU) %E
\ i
(D6) SQRT(2) SQRT(%PI) SIGMA > (-------------------------
/ 4
==== SQRT(2) SQRT(%PI) SIGMA
i = 1
2
(x - MU)
2
i (x - MU)
- ---------- i
2
----------
2 SIGMA 2
%E 2 SIGMA
- ------------------------) %E
2
SQRT(2) SQRT(%PI) SIGMA
(C8) expand(d7);
n n
==== ====
\ \ 2
2 ( > x ) MU > x
/ i / i
2 ==== ====
n MU n i = 1 i = 1
(D8) - ----- + ------ - --------------- + --------
SIGMA 3 3 3
SIGMA SIGMA SIGMA
(C9) solve([d8,d4],[mu,sigma]);
n n n
==== ==== ====
\ \ 2 \ 2
> x SQRT(( > x ) n - ( > x ) )
/ i / i / i
==== ==== ====
i = 1 i = 1 i = 1
(D9) [[MU = --------, SIGMA = - --------------------------------],
n n
n n n
==== ==== ====
\ \ 2 \ 2
> x SQRT(( > x ) n - ( > x ) )
/ i / i / i
==== ==== ====
i = 1 i = 1 i = 1
[MU = --------, SIGMA =
--------------------------------]]
n n
(C10)