# Poisson, Exponential problem

#### craig

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 $$\displaystyle Ex(12)$$?

#### Moo

MHF Hall of Honor
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

craig

#### craig

Cheers for that, was a typo in the original post.