If we know that:
How should I calculate the likelihood? I arrived to the next expression:
But something must be wrong because when I do this numerically, if n is big this likelihood will tend to 0... Easily deductible
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In practice, you would compute the logarithm of the likelihood, or loglikelihood. Then the numerical problems you mention should disappear. If the goal is to compute maximum likelihood estimates, then the loglikelihood is enough. Since $\log$ is a monotone increasing function, the likelihood and the loglikelihood is maximized at the same parameter values.
And then, if you want some indicator of the precision of those estimates, you will find that the hessian matrix of the loglikelihood at the estimate is useful, see https://en.wikipedia.org/wiki/Fisher_information or Theoretical motivation for using log-likelihood vs likelihood
If you want to choose between different models, then AIC (Akaike Information Criteria) is a possibility. Look it up, the loglikelihood again ... And soon you will discover that using loglikelihood is not merely convenient, it does also has many theoretical advantages.