# MLE of a Poisson distribution derivations

If we observe data $${{x_t}}^n_t$$ from the model, $$X_t sim Pois(tmu)$$ independently.

Then how would I derive the maximum likelihood estimate of parameter $$mu$$?

So far, I have done this:

Let $$X_t sim Pois(lambda_t)$$

Then $$f(x_t, tmu) = frac{tmu^{x_t}exp(-tmu)}{x_t!}$$

Thus I determined the Likelihood function to be

$$L(mu) = frac{tu sum_{t=1}^{n} x_t exp(-tmu)}{prod_{t=1}^{n}x_t}$$

So my next step would be finding the log liklihood. However I am confused as to what the notation for the log likelihood function would be. Would I be taking its as $$l(mu)$$ since $$mu$$ is the unknown parameter?

Thus I got $$l(mu)= log (tmu)sum_{t=1}^{n} x_t – ntmu +c$$

How would I derive the log likelihood function to give the MLE? Also would that mean that the expectation of the MLE would just be $$mu$$? Thank you in advance