Poisson Distribution - Earthquake Question??

In a particular region, earthquakes occur at an average rate of 3 per decade (one decade = 10 years), following a Poisson distribution. Find:

1) the probablity that the fourth earthquake after time t = 0 arrives within 2 years of the third earthquake

2) the probability that the fourth earthquake arrives by time t = 5 years

3) the probability that the elapsed time between the third earthquake to the tenth earthquake exceeds a decade?

4) the expected time and variance until the arrival of the next eight earthquakes

So what I have so far is: Let X be the number of earthquake occurences per decade so X~Poisson(3)

I am unsure of how to continue cause i am having difficulty with this topic. a little help to get me started on these would be greatly appreciated.

Re: Poisson Distribution - Earthquake Question??

Not my field, but...

You have a homogeneous Poisson process.

(*) The number of earthquakes within any time period of length T decades has Poisson distribution with parameter 3T.

(**) The interval (in decades) between any consecutive earthquakes has exponential distribution with parameter 3.

1) Use (**).

2) According to (*), the number N of quakes within a half-decade has Poisson distribution with parameter 1.5. P(N >= 4) = 1 - P(N = 0) - P(N = 1) - P(N = 2) - P(N = 3).