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so i have this question to answer

Assuming that ξ1 and ξ2 are two independent Poisson random variables with 
parameters, respectively, λ1>0 and λ2>0, lets define ξ3 as equal to ξ1+ξ2.
are ξ3  and ξ1 independent ? justify it.

since ξ3 follows a poisson distribution with parameter λ1+λ2 it should be indepedent right ? but i'm not sure and i'd like to know if anyone can help me prove it or tell me if it's false

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1 Answers1

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ξ1 and ξ3 are not independent. Proof: let's consider that ξ1 and ξ3 are independent. Then cov(ξ1,ξ3)=0 => cov(ξ1, ξ1+ ξ2) = 0 => cov(ξ1,ξ1) + cov(ξ1,ξ2) = 0. But we know that ξ1 and ξ2 are independent so cov(ξ1,ξ2) = 0 and what we get is: cov(ξ1,ξ1) = 0 => var(ξ1) = 0 which doesn't hold because var(ξ1) = λ1 > 0.

al.al.
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    Would it not be easier to write $\text{cov}(ξ1,ξ3)= \text{cov}(\xi_1, \xi_1+ \xi_2) = \text{cov}(\xi_1,\xi_1) +\text{cov}(\xi_1,\xi_2) = \lambda_1>0$ or something similar, and avoid the need for a contradiction? – Henry Jan 31 '18 at 00:14