1 I will give an intuitive explanation, only needing the defining properties of the Poisson Process and the exponential distribution, without needing any calculations involving densities. In short, the Poisson Process having independent and stationary increments and the exponential distribution being memoryless explains this connection:
If two events occur at a rate of 1.8 per hour on average, and this occurrence follows a poisson process, what is the probability that there is at least 1 hour between two events? My approach for t...
Finding the probability of time between two events for a poisson process
The criteria in your book are for interval data; that would be useful if you had the dates at which hurricanes stroke... moreover these criteria are for constant rate Poisson processes, which is obviously (or I hope so) not the case of hurricanes. To check if your count data follow a Poisson distribution, a first elementary approach is the chi-square test.
How to know if a data follows a Poisson Distribution in R?
The number of cars appearing in a car park follows a Poisson distribution with a mean of 10 cars per hour. Find the probability of there being: (a) Exactly 5 cars in a 30 minute interval, (b) At most 3 cars in a 10 minute interval, (c) More than 4 cars in a 15 minute interval, (d) Exactly 1 cars in each of three consecutive 5 minute intervals.
As alluded to, this formula for P is found by composing the "nice" known solution for the Poisson kernel in the unit disk $\mathbb {D}$ through appropriate biholomorphisms so that it can be written as a function from $\tilde {A}$.