Let $(X_t)_{0 \leq t \leq 1}$ be a Poisson process with intensity parameter $\lambda > 0$. The process is observed in $k$ time points $0 \leq t_1 < \cdots < t_k \leq 1$. Determine the MLE for $\lambda$.
I understand the task in the exercise that there exist values $x_1, \dots, x_k$ such that that $N_{t_i} = x_i$ is observed and thus using the fact that $$P(X_{t_i} = x_i) = \frac{(\lambda t_i)^{x_i}}{x_i!}e^{-\lambda t_i}$$ it follows that: $$L(\lambda;x_1,\dots,x_k) = \prod_{i = 1}^k \frac{(\lambda t_i)^{x_i}}{x_i!}e^{-\lambda t_i}$$
After applying the logarithm and deriving the function, I get as MLE for $ \lambda$: $\hat{\lambda} = \dfrac{\sum_{i=1}^k x_i}{\sum_{i=1}^k t_i}$
Is that correct? I have a feeling that I misunderstood the assignment itself.