My question is in two parts:
1- if I have a probability that an event would occur after one year, how
can I compute this probability after 6 months and after 2 years?
2- Are the computations above going to change if I consider the
probability of occurence discrete or continuous? if so how can I account
for the fact that the event is discrete or continuous
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What you’re looking at is a Poisson distribution. This assumes that the
event occurs randomly but at a constant “average” rate.
For this problem it is easier to calculate the probability that the
event has NOT occurred after a certain time span t. This is given by
P_not_occurring(t) = e\^(-lambda * t)
where t would be measured in years for your problem, and lambda is a
P_occurring(t) = 1 - e\^(-lambda * t).
This measures the probability of the event occurring at least once
within the time span t.
You can calculate the value of lambda if you have the probability of the
event occurring after one year, by substituting 1 in for t:
P_occuring(1) = 1 - e\^(-lambda)
e\^(-lambda) = 1 - P_occuring(1)
lambda = -ln(1 - P_occuring(1))
where ln is the natural logarithm.
In this case the probability of occurrence is always continuous
(assuming that the event truly is Poisson distributed). I’m not sure
what you mean by saying that it might be discrete.
You could just make a line graph and average the slope.