Maximum of poisson process

Question

Let $X^{(1)}_{t\ge 0},...,X^{(n)}_{t\ge 0}$ independent Poisson Processes with common intensity $\lambda$

Find the distribution of the first time that

a)at least one event has ocurred in every process

b) the first event occurs in any of the process

For the first part I compute: $$P[X^{(1)}_{t\ge 0}\ge 1,...,X^{(n)}_{t\ge 0}\ge 1]=(P[X^{(1)}_{t\ge 0}\ge 1])^n=(1-P[X^{(1)}_{t\ge 0}=0])^n=(1-e^{-\lambda t})^n$$ so $F(t)=(1-e^{-\lambda t})^n$ but I don´t know if this is correct

And for the second part I don´t know hot to do it. Do I need to compute the maximum of the process?

I would realy appreciate if you can help me with this problem

Answer 1

Hint: the sum of independent Poissons is Poisson...