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let t1 be the first arrival time which van be computed from min{X1,...,X5}

t2 is the time of the next and so on. and assuming independence of arrivals and each is ~exp(lambda)

suppose we have the data:

day 0, 1, 2, 3

# arrivals 0, 2, 3,5

means at day 0 there are 0 arrivals, day 1 there 2 arrivals,..., day 3 total arrivals 5.

if we let T=min{X1,...,X5} then the minimum is lower bounded by t1:

P(T>=t)= P(min{X1,...,X5}>=t1)=P(X1>=t1).P(X2>=t2)...P(X3>=t5)=exp(-5*lambda*t1)

and our cdf is: F(t1)=1-exp(-5*lambda*t1)

so we can determine the arrival time of any,

my question is how to compute the probability of observing c people?

for example P(observing 2 persons)?