Sir, Please explain how the time to the first claim in a poisson process has an exponential distribution with parameter lambda. Also please explain how the time between claims also has exponential distribution. I don't understand how the derivation. Thank you . Vaishno
The long derivation are unlikely to be asked. However there is a shorter proof earlier on in the chapter which works well. If waiting time is given by T, then: P(T>t) = P(0 events up to time t) = P(0 events in a \(Poi(\lambda t)) = e^{\lambda t}\) So: \(F(t) = P(T \leq t) = 1 - e^{\lambda t} \) Hence: \(f(t) = F'(t) = \lambda e^{\lambda t} \) this is the PDF of an \(Exp(\lambda)\) - hence the waiting time has this distribution. Due to the memoryless property of the exponential - the waiting time between events will be the same.
