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Just a quick question on MLE, sorry if its too basic but I would love help on this issue!

If we want to calculate the probability $P$ of the number of males in the United States and we knew the distribution $pmf$/$pdf$, why can't we just take the arithmetic mean and use $\bar{y}$. Why go through the trouble of using the MLE?

For example, if we suppose that we are looking at a Bernoulli R.V. where $X_1=1$ is male and $X_2=0$ is a female and we want to estimate the parameter, which is $p$ in the case of Bernoulli, why would we go to great lengths to find the MLE when we could just see the $mean$ from our sample?

I'm very confused on why MLE is useful in that way especially when we have the data. Thanks so much!

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up vote 4 down vote accepted

MLE gives you the most likely answer, i.e the most probable answer. MLE is not always the "best" answer but in many cases, it happens to be the best.

To answer your second question, your intuition of finding the mean happen to be right but you have to be careful about using the mean as the only way to solve a problem. Let me give you an example where the mean is not a reliable way.

Say you have a RV that is uniform in $[0,A]$ and you want to estimate $A$. Suppose that we take four sample values and they are $1$, $2$, $3$ and $34$. The mean of your sample is $10$. We also know that uniform RV in $[0,A]$ has a mean of $A/2$. So blindly trusting the mean will lead you to believe that $A=20$ which is absurd because we have the sample $34$. So mean is not a good indication at all.

It turns out that $MLE$ is not a good estimate also! If $A<34$, then the probability that we will see $34$ is zero. If $A>34$ then the probability that all the samples are less than $34$ decreases as $A$ increases. So the most probable value (MLE) is $A=34$. This turns out to underestimate $A$!

The above problem has a very interesting history. During world war II, the British knew that Germans would assign serial numbers to the tanks. So the serial numbers come from a uniform distribution between $1$ and $A$ where $A$ is the number produced. So based on the tanks that were captured/destroyed, the British had random samples from the uniform distribution. They used the unbiased estimate of $A$ to get very accurate estimate of the number of tanks in the German army. Look at

To cut a long story short, MLE is not always the best estimate, except in the absence of any other facts, it is the best we got

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