Solving a two equation system yields strange results

Yes, this particular equation is only defined if k is an integer.
However, even after I make this known in maxima, the system appears to
have no solutions:

(%i2) declare(k, integer);
(%o2)                                done
(%i3) solve([b*(k - 1) = 0.1, exp(-1/b) * sum(1/(i! * b^i), i, 0, k
-1) = 0.02], [b, k]);

rat: replaced -0.1 by -1/10 = -0.1

rat: replaced -0.02 by -1/50 = -0.02
(%o3)                                 []
(%i4) solve([b*(k - 1) = a, exp(-1/b) * sum(1/(i! * b^i), i, 0, k -1)
= 0.02], [b, k]);

rat: replaced -0.02 by -1/50 = -0.02
(%o4)                                 []

FYI, what I'm trying to do here is to fit the parameters ? (theta) and
k of a gamma distribution such that 98% of its mass is contained
within the interval [0, 1]. In addition, I require that the "top" of
the distribution, which is situated at ?*(? - 1) is equal to 0.1, or a
in the general case. Unfortunately I can't use the first
characterization of the CDF (as described in the link above, where x =
1) to do this because Maxima does not support the incomplete gamma
function ?), so I'm using the second characterisation, which includes
a sum of k terms.

Two unknown quantities, two equations; there should be a solution to this.

Jason

On Wed, May 9, 2012 at 11:24 PM, Raymond Toy <toy.raymond at gmail.com> wrote:
>
>
> On Wed, May 9, 2012 at 11:46 AM, Jason Filippou <jason.filippou at gmail.com>
> wrote:
>>
>> Good afternoon.
>>
>> I've been using ?Maxima 5.21.1 in a Debian GNU / Linux 2.6.32-5-686
>> system to solve a particular two equation system that I have to. The
>> system is as follows:
>>
>> (1): b*(k - 1) = 0.1
>>
>> (2): exp(-1/b) / sum(i! * b^i, i, 0, k -1) = 0.02
>>
>> I've been using solve/2 for this, but I've been returned an empty
>> solution set. Namely:
>>
>> %i9) solve([b * (k - 1) = 0.1, exp(-1/b) / sum(i! * b^i, i, 0, k -1) =
>> 0.02], [b, k]);
>>
>> rat: replaced -0.1 by -1/10 = -0.1
>>
>> rat: replaced -0.02 by -1/50 = -0.02
>> (%o9) ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? []
>>
>> Now, normally I would assume that this means that the system doesn't
>> have a solution, but after substituting beta with its equivalent from
>> the first equation, i.e 0.1 / (k - 1), I noticed that the evaluation
>> of the second equation halts after a couple of steps:
>>
>> (%i5) solve([exp(-(k - 1) / 0.1) * sum(1/(i! * (0.1 / (k-1))^i), i, 0,
>> k -1) = 0.02], [k]);
>>
> This seems a bit ill-defined.? Since k is the upper limit of the sum, what
> do you expect sum(...,i,0,k-1) be when k is not an integer?? Do you mean to
> take the floor of k-1?
>
> Ray
>



-- 
Jason Filippou
Research Associate
NCSR Demokritos
NCSR Webpage: http://users.iit.demokritos.gr/~jfilip/
D.I.T Webpage: http://cgi.di.uoa.gr/~std06142/
LinkedIn! Profile: http://www.linkedin.com/profile/view?id=132927442&trk=tab_pro