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We have a queue where people pass out of it with $ \text{Poisson}(\lambda) $ and they come in with probability $ p $.

I understand that the arrivals follow $ \text{Poisson}(\lambda p) $, but how can I prove it?

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I dont understand your question the way its worded. It sounds like you want us to prove a given? – CogitoErgoCogitoSum Feb 14 '13 at 3:35
What do you mean "come in with probability $p$"? Also, I don't understand how arrivals follow Poisson distribution with smaller mean than departures unless something funny is happening in the queue. – Inquest Feb 14 '13 at 5:04
@Inquest +1 for something funny. – Did Feb 14 '13 at 6:56
I mean that people decide if they are coming in with probability p,otherwise they pass outside of it and they are not been counted as arrivals.Only for those who come in the mean of arrivals and departures is going to be the same,λp. – user52561 Feb 14 '13 at 10:32

This is called thinning a Poisson process. These Applied Mathematics notes elegantly describe the situation and prove that the thinned process is indeed Poisson.

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