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Since we can compute the mean and the standard deviation of a set of random variables, why do we use Maximum Likelihood Estimation to estimate these parameters?

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  • $\begingroup$ The MLE of a normal RV consists, indeed, in the empirical mean and variance. Where did you encounter the use of MLE to estimate a normal distribution parameters? $\endgroup$ Nov 12, 2019 at 19:55
  • $\begingroup$ @RomainReboulleau In all materials I checked. See Theodoridis, S., & Koutroumbas, K. “Pattern recognition. ” Fourth Edition, 9781597492720, 2008 $\endgroup$
    – Yacine
    Nov 12, 2019 at 20:00

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While it is true that we can compute the sample mean and sample standard deviation from a set of random variables, the Maximum Likelihood Estimation provides a formal framework for estimating the population parameters based on observed data.

In most cases, we are more interested in what we can imply for the population than what we know about the sample, this is why we use the estimation, since the sample usually has some flaws in how it is created.

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