Decomposition de probabilité exponenetielle

[saram123]

Hello,
I have two Poisson processes with parameters Lambda1 for the first and Lambda2 for the second. this give a Poisson process with parameter the sum (Lambda1+lambda2). so, the inter-arrival is exponential probability with parameter the sum ((Lambda1+lambda2).
I want to decompose the exponential function in two function f1, f2 depending on lambda.
How can I do it?
I tried with the total probability formula. But , I made an error as I suppose that f1(lamda1)= lambda1*exponential(-labmda1*x). However it is false.
How can I calculate f1 and f2. All information that I have that each process have probability p (resp (1-p)) to happen.
Any suggestion could help me to solve this please. :id:
Thanks.

[Ben314]

First, excuse me for my poor english : as lots of french peoples, I only speak (badly) french...

I'am not sure you can decompose the exponential function in TWO function :
When you count the interarrival time between two "events" for your two Poisson processes P1 and P2, there is four case for this interarrival time :
- It's the time between two event of the P1 process (whithout event of the P2 process betwen them)
- It's the time between two event of the P2 process.
- It's the time between an event from the P1 process and an event from the P2 process.
- It's the time between an event from the P2 process and an event from the P1 process.
...

[saram123]

First, excuse me for my poor english : as lots of french peoples, I only speak (badly) french...

I'am not sure you can decompose the exponential function in TWO function :
When you count the interarrival time between two "events" for your two Poisson processes P1 and P2, there is four case for this interarrival time :
- It's the time between two event of the P1 process (whithout event of the P2 process betwen them)
- It's the time between two event of the P2 process.
- It's the time between an event from the P1 process and an event from the P2 process.
- It's the time between an event from the P2 process and an event from the P1 process.
...

re,
Merci pour ta réponse, pas de problème pour la langue vu que je comprend le français.
Je doit absolument faire distinction entre les deux type d'arrivé.
Je ne sais pas trop ce que je peux utiliser. Je vais encore chercher. :mur: