Reference for 1-exp(-t/T)

katxt

Active Member
#1
Hi. I'm looking for a quotable reference that looks something like Pr(at least one occurrence in time t | average waiting time = T) = 1-exp(-t/T) for events happening at random. Everywhere I look I find rate version 1-exp(-lambda.t) which is identical, but not the version I want.
Any clues? Cheers, kat
 
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#2
i am not sure if that may be what you mean, but here is what i figure it out, hope i can contribute in some way

always P(at least one) = 1 - P(0)

example:
in ww2 to hit an aeroplane, chance per rifle = 0.03
but what if a group of 150 infantry shots at the same time -> then 0 hits = 1.04% and at least 1 = 98.96%
( 1-0.03)^150 = 0.0103, 0 hits -> at least one hit =1-0.0103=0.9896)

1/lambda is suppose to be the mean
P(0)=lambda
by average 1/lambda

then P(at least one) = 1-P(0)

(assuming P(X) = lambda x exp(-lambda x X) if X>=0)

i not understand what version u say
 

katxt

Active Member
#3
Thanks, mvernengo.

My version is this - you start walking through a forest looking for a particular sort of bird. Assume we know from past surveys that the average time to see this sort bird for the first time is T = 30 minutes. We search for t = 20 minutes. The probability that we see at least one bird of this sort in the 20 minutes given that the average is 30 min is P = 1 - exp(-t/T) = 1 - exp(-20/30) = 49% approx. This is based on the Poisson/exponential theory that you have used in your post.

However, I want to use this formula in a paper I am writing for biologists and I would like to quote the formula Pr(at least one occurrence in time t | average waiting time = T) = 1-exp(-t/T) as a known result rather than confuse the issue by having to derive it in the paper. So what I am looking for is a book or something which I can quote to save the readers time.
kat
 
#4
After much scouring through books and journals, the closest thing I was able to find is on this page—look for the heading “6.3 Poisson process” where it says:
A Poisson stochastic process has the property that events are independent, and the interarrival times of events can be described using the exponential distribution F(t) = 1 – exp(−λt). … Given that the mean time between some event is 1/λ, the rate of occurrence of the events will be λ.


My suggestion would be to write it as 1 – exp(–t/T)” in the body of the paper, and add a footnote with a reference to the above citation, saying something like “Here we let λ = 1/T for convenience.”

That way, youll introduce it into the literature yourself as a citable reference, assuming your paper gets accepted for publication.
 
#5
yes, totally a poisson process
mu (mean) = lambda x t
so in this case lambda = 30 and t=2\3
the only variable r=1
Pr(r=1)= mu^r x exp(-mu) | r!
 

katxt

Active Member
#6
Thank you both, and an excellent suggestion Con-Tester. I'm surprised that such a useful parameterization isn't widely used. kat