Can you help me find a generally recognised mathematical rule/law/curve which most closely matches a fixed set of numbers?

[Jamie C]

Hi, I hope someone can help me with this problem! I have a distribution of 2000 numbers but I only know the first 10. The first 10 numbers are: 
2025, 1000, 335, 300, 187.5, 135, 99.5, 20, 17.5, and 13.5. I know that the total of all 2000 numbers is about 12,000. 

What I am looking for is a recognised long tail curve model which using the first 10 numbers only predicts a total for all 2000 numbers in the sequence. I need the total to ideally be between 11,000 and 13,000. I did look at Zipfs law but I'm not sure this works. 

I am not a mathematician but would appreciate any recognised models/distribution curves/laws you could suggest and what the model you suggest would give as a total for all 2000 numbers in the sequence knowing only the first 10 numbers. 

The key thing that I am trying to do is fit these numbers to a recognised curve / long tail model. I am less bothered about the total (although ideally in the range 11000-13000) and more bothered about the model being something that would be recognised by mathematicians globally.

I hope this is clear. Thank you so much in advance to anyone who tries to solve this for me!  

[OldChemE]

You haven't said whether these numbers are the result of a specific calculation, or whether they are experimental observations.  They very much appear to be exponential decay, but not all the numbers fit perfectly.  My scientific calculator suggests that a reasonable mathematical model (assuming exponential decay) could be y = (2870)*(0.576)^x, where x =1 for the first term (the 2025 term).  Exponential decay does have a long tail as you suggested.

[Jamie C]

Hi, Thanks for your reply. To give you the context the numbers are actually costs. I know that there were approximately 2000 costs incurred. I don't know the exact total spent but believe it was approximately $12,000 ...most of the costs were just small amounts. 

Lots of people are estimating the total cost at $12,000 (we will never know for sure the total cost as we can't find out what every single one of the 2000 costs incurred actually were). I want to see if I can add a mathematical dimension to all the people just sticking their finger in the air and guessing what the total is. 

What I'm trying to get to is a statement that says something like "If the 10 known costs know follow an XYZ distribution then the total costs incurred would be XXXXXX" where XXXXX is a total cost of around $11,000 to $13,000. I don't know if this is even possible but hopefully someone can pin this to a recognized distribution/rule/law/model/curve. 

Hope this explains! And thanks again for your reply. Apologies I am not in any way an expert in maths so really appreciate any help I can get ! 

Jamie. 

[Bufofrog]

The 10 costs you showed add up to about about 33% of the total of $12,000.  But the 10 costs represent only .5% of the total number of costs.  That means to me that your 10 cost sample is not representative of the total.

I do not see how you could get anything that is useful out of these numbers since your sample is not representative of the total.  I am not a statistician so maybe there is some "magic" they could do to give you something useful.

Good luck. 

Edited May 8, 2019 by Bufofrog