Need help with an interesting limit
Greetings all! Here's an interesting problem I could use advice & council on. I'm attaching a short Word file with equations and a picture.
I have some aircraft accident data, which are histograms of accident probabilities given pilot flight experience (x-axis), so y-axis 0<=p<=1.
I fit a gamma pdf (with a base-rate term added) to these data.
Now, given each fitting function y=f[x], I'd like to find a function describing the probability of NOT having an accident over a known range of x.
To begin, f[x]= b + (A e-(Log[x]-delta)/beta (Log[x]-delta)alpha-1 beta-alpha) /Gamma[alpha], which is a base rate (b) topped by a gamma pdf containing an amplitude term A and a location (shift) parameter delta. Mathematica then gave me values for all the parameters of f[x].
Next, given the histogram nature of the data, if the probability of having an accident at x is f[x] over the range of that bin, then the probability of NOT having an accident at x should be 1-f[x].
However, the height of f[x] itself represents a bin value 100 units wide (accident pilots/(accident pilots+no-accident pilots) at that value of x. So, let's call the width of each bin dx, which starts at 100.
Then, the probability of having NO accident over the span of, say, x1 to x2, should be a definite product. Each term can be represented as (1-.01 dx f[x]), where .01 corrects for the original bin width (dx=100).
Given individual bins starting at (1-.01 dx f[x]), the probability of having NO accident over the range x1 to x2 should then be the definite product (Cap-pi, see attachment, i.e., all those terms multiplied by each other). Finally, the probability of having AT LEAST ONE accident over x1 to x2 should then be 1-this definite product.
It occurred to me that there should be a limit to that definite product, so I checked it numerically by shrinking dx toward zero. And, yes, the definite product stabilizes as dx gets smaller.
But, I'm having trouble finding the actual limit itself. Mathematica chews on it, then gives up. It's certainly non-trivial, since, each time dx halves, the number of terms in the definite product doubles at the same time each individual term is trending toward 1.
So, this is a definitely interesting problem, but admittedly beyond my ability. I tried logarithms (i.e., the log of the definite product to transform it into a sum). But, the fundamental logical problem remains, namely that the number of terms increases as the value of each decreases.
What trick could be applied to this to solve it?