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I get using Maximum Likelihood Estimation to find unknown parameters of a function.

But in the normal distribution, we know probability density function is f(x)=1/σ√2π(e^−(x−μ)2/(2σ^2)) where μ is mean of our distribution and σ is the standard deviation.

Here, we already know the formulas of mean μ (sum of observations/total observations) and standard deviation SD (∑(√∣x−μ∣^2)/N) i.e. formulas which are fixed and depend on values in our data. Why then would we use MLE to find μ and σ? Why are μ and σ parameters that need to be estimated? (Like in here: https://www.statlect.com/fundamentals-of-statistics/normal-distribution-maximum-likelihood)

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    $\begingroup$ " the formulas of mean μ (sum of observations/total observations)" ... no that's the sample mean $\bar{x}$, not the population mean, $\mu$. Similarly with the standard deviation. You can't calculate the population values without observing the whole population. $\endgroup$
    – Glen_b
    Commented Aug 21, 2021 at 7:22
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    $\begingroup$ You know the formula for the sample mean is $\bar x =\frac1n \sum x_i$ but due to randomness in sampling this is unlikely to be equal to $\mu$, just an estimator of $\mu$ which happens to be unbiased and for a normal distribution the maximum likelihood estimator and minimises the expected mean-square error. But for estimating the variance, no single estimator has all three properties: one possibility is $\frac1{n-1} \sum(x_i-\bar x)^2$, another is $\frac1{n} \sum(x_i-\bar x)^2$ and a third is $\frac1{n+1} \sum(x_i-\bar x)^2$ and there are potentially others $\endgroup$
    – Henry
    Commented Aug 21, 2021 at 7:27

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