Consider waiting for an event for a time t in a Poisson process with on average a arrivals per time unit.

a$)$ By considering P$($N$($t$)=0),$ argue that P$($waiting time is $>$ t$)=$ exp$(-$at$).$

b$)$ Using your result in a$),$ determine the probability density of the waiting time between successive arrivals in a Poisson process with on average a arrivals per time unit.