Poisson Process

[jjangub]

I got a question.
Q) Let {X(t), t>=0} and {Y(t), t>=0} be independent Poisson processes with parameter and respectively. Define = X(t) + Y(t) and = X(t) - Y(t) and = X(t) + k, k a positive integer. Determine which of the above are Poisson and find lambda.

so I figured that is limiting case to the Binomial distribution and the lambda is ()/( + )

and for , it is Skellam distribution and lambda is -

and for , even though you add positive integer k to poisson process, you get + k.

I not sure that this is right...
I got the idea from the internet (except for the last one).
and for , there is no lambda?
b/c Binomial distribution doesn't have mean.
Please tell me what I did wrong.
Thank you....

[vinux]

Are you sure about the first one. What is the distribution of sum of two independent Poisson random variables?

[jjangub]

I actually found that the mean of binomial distribution is number of trial * success probability in each trial, so this is lambda...
how about Z2 and Z3? are they okay?