Estimating frequencies of a population I have a sample of size 1121 from a population of size 2171 and I don't have access to any additional samples. The counts are
   1   2   3   4   5   6   7   8   9  10  11  12  13  14 
 737 158  70  44  24  15  20  12   6   9   3  10  10   3 

I am trying to estimate the frequencies for the  population,  
All points in the population are positive.  Would the naïve estimate of 
$(737/1121)*2171$ points have value 1, 
$(158/1121)*2171$ points have value 2, ect... 
be the appropriate measure. 
The sample is count data in a capture-reacpture analysis, i.e., among all capture periods there were a total of 1121 individuals captured. Among the captures, 748 individuals were captured once, 158 twice, ...these captures were actually buyers of a product. This was only small a portion of the total buyers. The total buyers are the population that is being estimated. It is assumed that the buyers that were not observed exhibit the same behavior as the buyers that were captured. I want to get a good estimate on the "buying" counts on the estimated population for each individual.
Would fitting a distribution be the right approach? If the distribution is the appropriate way, would I generate 2171 random values and use those counts to get my estimate.
 A: Without more details, I find it risky to fit a distribution. At least for the first categories, you can have fairly good estimates for your population. 
The value you gave is correct, but let's have a look on the 95% confidence interval for the first categories. 
$$\hat{p}\pm z_{\alpha/2}\sqrt{\dfrac{\hat{p}(1-\hat{p})}{n} \cdot \frac{N-n}{N-1}}$$
You have $z_{\alpha/2}=1.96$, $N=2171$, $n=1121$  . According to my calculations it would give this simplified formula : $$\hat{p}\pm 0.04071\times\sqrt{\hat{p}(1-\hat{p})}$$
The highest width of this interval is given by $\hat{p}=1/2$ so you will end up with a 95% confidence interval more narrow than 0.02.
After your comment "So technically it is not a "sample" from the population" I am not really sure whether it's a legit estimation. Maybe you should go into more details how this sample is produced and why you want this estimation. 

Edit after OP details
As I really don't know how to treat capture-recapture analysis, maybe there are some related distributions that indeed fit. I would suggest to edit your question adding these new informations. But, at first glance I don't see a problem with my answer. However as the counts should theoretically go down, maybe only fitting 6th to 14th with a distribution even a very simple one (like linear) could grant better estimations than what I suggested. But ...at second glance...this comment also suggest that the data can be really noisy, and that you want to predict future outcomes by this recapturing process. This is really not the same thing as sampling measures from a population. 
So, finally, if what I think about your data and your goal is right, my answer can't really hold and I would suggest to go to fit your data with a distribution.
