Let $\{N_t\}$ be a Poisson process that refers to births in month February (28 days), of rate $\lambda>0.$ Given that in February we had 28 births, find the distribution of the number of births that took place on Mondays (total $4$ of them).
Attempt. Given that $N_{28}=14,$ the distribution of $N_4$ is Binomial$(14,4/28)$. But how are we sure that $N_4$ refers on Mondays and not on 4 random days for example?
To fix this, given $0\leq k\leq 28$, I thought on working with $$N_1=k_1,~N_8-N_7=k_2,~N_{15}-N_{14}=k_3$$ and $N_{22}-N_{21}=k_4$ for all possible combinations of $k_i$'s such that $k_1+\ldots+k_4=k$. Since $N_1,~N_8-N_7,~N_{15}-N_{14},~N_{22}-N_{21}$ are independent rv's from the Poisson($\lambda$) the calculation would go like:
$$\sum_{k_1+\ldots+k_4=k}\frac{ e^{-\lambda}\frac{\lambda^{k_1}}{k_1!}\ldots e^{-\lambda}\frac{\lambda^{k_4}}{k_4!} }{ e^{-\lambda}\frac{\lambda^{k}}{k!} }$$
Is the first or the second approach correct?