Poisson process - time to the first claim & Time between Claims

[vaishno devi makam]

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

 

[John Lee]

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.

 

[vaishno devi makam]

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.

Thank you sir!