Let $N, Y_n,n\in\mathbb{N}$ be independent random variables, with $N \sim P(\lambda), \lambda \lt \infty$ and $\mathbb{P}(Y_n = j)=p_j$, for $j=1,\dots,k$ and all $n$. Set $N_j = \sum_{n=1}^{N}\mathbb{1}(Y_n=j).$ Show that $N_1,\dots,N_k$ are independent random variables with $N_j \sim P(\lambda p_j)$ for all $j$.
For distribution I used that $N_j | N \sim Bin(N, p_j)$ and so $\mathbb{P}(N_j=k)=\sum_{N=k}^{\infty}{{N}\choose{k}}p_j^k(1-p_j)^k\frac{\lambda^N}{N!}e^{-\lambda}=\dots=\frac{(p_j\lambda)^k}{k!}e^{-\lambda p_j}$ Need help with independence.