So I have a question relating to how the negative exponential distribution arises when talking about probability, more specifically, when deriving a probability from a rate. I believe it involves concepts related to the hazard function, as well as Taylor series expansions. Hopefully someone can recognize what I'm trying to say and point out what assumptions I am forgetting or what concepts I need to know to better articulate the problem and solution. (Please excuse my fast and loose 'verbal' math).(adsbygoogle = window.adsbygoogle || []).push({});

My question is thus:

so we have a an initial amount k_{0}and we wish to calculate the new amount k_{1}after time t. the amount change is equal to the initial amount times the rate of change over the interval, λt. this is a stochastic process.

so we have

k_{1}= k_{0}- k_{0}*λt

k_{1}= k_{0}(1 - λt)

So the amount change is λt*k_{0}. Another way of putting this is that the amount change is equal to (k_{0}* the rate at which the event occurs once per time period t * time interval), plus the rate at which the event occurs twice per time interval, which is equal to the the first term squared (since the rate at which a second event occurs is a function of the rate a first event occurs), and the rate at which a third event occurs is equal to the first two terms multiplied by the first term, and so on. Now if we shrink t towards zero, dividing by a factor equivalent to the number of terms in the series [what says we have to do that btw??], the higher powers of the first term in the above series drop out, and so the event occurs once during the interval or not at all. The proportion of change λt then equals the probability of the event occurring, and the complement 1-λt is the complement of this probability.

Now using the geometric distribution, we can calculate the probability of the event occuring in the n+1th interval t after n intervals of failure, (λt)(1-λt)^{n[\sup], after which by substituting a variable and applying the limit (1-1/n)^n as n-> ∞ =e we get e-λt[\sup]. [errors?] So a few questions here. One, not sure if I quite understand what I'm saying :) about the rate at which the event occurs once per time interval; does this necessarily mean that the amount would necessarily be less than 1? Or if it equals 1, then the series wouldn't converge. Also, do the rates for the 'higher powers' of the first term of the series necessarily have to be the same, at least if the initial rate is not the instantaneous rate? If you understand what I am trying to say and can point out the mathematical concepts I need to get there, I'd greatly appreciate it. Am I mismatching concepts from different definitions in ways that don't make sense? (and btw, I had to rewrite this when physicsforums signed me out initially but I don't think I left anything out!)}

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# Hazard Function, and how to 'turn a rate into a probability'

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