Poisson, Exponential problem

Apr 2008
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159
Little stuck on a Poisson problem.

Sightings of a particular species of rare bird occur at a rate of 6 per year. Assume that sightings can be modelled by a Poisson process \(\displaystyle (N(t), t>0)\) where time \(\displaystyle t\) is measured in years and time 0 is the present time.

Let \(\displaystyle T_1\) be the time from time 0 until the 1st sighting. What is the probability that \(\displaystyle T_1\) is at most:

i) 1.
ii) 2.
I remember something about the exponential distribution being the time between Poisson processes, but not sure how to apply it here.

Would \(\displaystyle T_1\) be distributed \(\displaystyle Ex(12)\)?

Thanks in advance
 

Moo

MHF Hall of Honor
Mar 2008
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P(I'm here)=1/3, P(I'm there)=t+1/3
Hello,

Why 12 ? I think the Ti follow an exponential(lambda), and here, lambda=6

P(T1<1)=1-P(T1>1)=1-P(N(1)=0), because T1 is the time of first sighting. So if it's >1, it means that between 0 and 1, there have been no sighting.

And N(1) follows a Poisson distribution with parameter lambda.
So P(T1<1)=1-exp(-lambda)

Similarly, P(T1<2)=1-P(N(2)=0), and N(2) follows a Poisson distribution with parameter 2*lambda
 
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Apr 2008
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Cheers for that, was a typo in the original post.