is the sum of two poisson distribution independent from the others variables

Question

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

Answer 1

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.