• We are pleased to announce that the winner of our Feedback Prize Draw for the Winter 2024-25 session and winning £150 of gift vouchers is Zhao Liang Tay. Congratulations to Zhao Liang. If you fancy winning £150 worth of gift vouchers (from a major UK store) for the Summer 2025 exam sitting for just a few minutes of your time throughout the session, please see our website at https://www.acted.co.uk/further-info.html?pat=feedback#feedback-prize for more information on how you can make sure your name is included in the draw at the end of the session.
  • Please be advised that the SP1, SP5 and SP7 X1 deadline is the 14th July and not the 17th June as first stated. Please accept out apologies for any confusion caused.

CS2 ch1 Q1.4(ii) when compound poisson process is also a poisson process

I

iamminime

Active Member
ch1 Q1.4(ii) it states that
when St = X1 + X2 + ... + XNt is a compound Poisson process, if Xj can only take value 0 or 1, then St is also Poisson process.

I can understand when Xj = 1 for all j, St is a Poisson Process, because all Xj then sum up to Nt, which is a Poisson Pcoess.
But why when some of Xj = 0, St is also Poisson Process? Thanks a lot!
 
Hi Iamminime

Good question! This is similar to the idea of thinning a Poisson process. Say we have that Xj takes the value 0 with probability 50% and 1 with probability 50%. The process St = X1 + X2 + ... + XNt where Nt ~ Pois(lambda * t) is the same as a Poisson process with parameter lambda / 2, ie it is the same as the following process:

Rt = Y1 + Y2 + ... + YMt
where Yj = 1 with certainty and Mt ~ Pois(lambda / 2 * t)

Hope that helps!

Andy
 
You must log in or register to reply here.
Back
Top