I have found lots of problems that I am working on that mimic this pattern.
The pattern as I see is (using my question as an example): age is always positive integer
Right tail after certain point is all 0 (for this example, any number before 14 is always zero)
Short exponential on the right(a quick growth period until...)
A period of plateau of a large jump (this is the 4 or 5 year cohort in my example)
A long dropoff with an eventual extinction (small decline until extinction around 70)
So I see a few parameters here: MinThreshold,Rate of exponential incline, Peak, MaxThreshold, Rate of exponential decline
I can multiple the whole equation by (using excel for prototype)
=IF(ageOfUser>peakAge,0,IF(ageOfUsermaxThreshold,0,userAge))) * slow exponential equation or Fibianci with loss
added to
=IF(ageOfUser<=maximumThreshold,0,IF(ageofUser > maximumThreshold,0,ageofUser)) * fast exponential decrease equation
Depending on the data set, there might be a plateau period which I would need to define. I would think this type of equation would explain population over time for species also.
So two questions, 1> How the heck would I represent this in mathematical equation?
2> Has anyone already done something in biology for an equation like this? (I realize this might not be the exact forum but I am hoping for a mathematical biologist might see this)
Lulus answer to use log-normal can give me the flavor of what I need. Setting mu =0, and sd =1 gives the curve.
So I have in excel:
=LOGNORM.DIST(YearsPastMin,0,1,FALSE)
I then use a modifier to get it to peak at the correct place (In this case 18%)
=LOGNORM.DIST(YearsPastMin,0,1,FALSE)* modifier(3.4) * n
Interesting enough, the squared error is actually the same curve with certain params but it still has too much error.
So while I still have another question, this is the equation I was looking for. I need to figure out how to manipulate the slope for both sides but that will have to wait until another question.