Why need to estimate the distribution parameter through MLE when they can calculated them from observation? MLE is used to estimate parameters of distribution that can best fit our observed data. However, why do we go through these trouble since the parameters can be directly calculated from observed data? For example, to estimate the mean of a Normal distribution, we simply calculate the mean of the sample.
 A: Very often, there are several possible (and reasonable) ways to estimate parameters from data. MLE gives us a sensible way to find one of them with some good properties.
I suppose the question comes from the fact that often the ML estimator is just the estimator anybody would have obviously used without all the trouble of finding the MLE. For example, even without knowing what likelihood is, anybody would rightly guess that the most reasonable way to estimate the mean of a normal distribution is to take the mean of the sample - in fact, our usual trouble is to make students tell apart sample mean from population mean.
However, not all cases are so obvious, because there isn't a self evident parallelism between sample statistics and distribution parameters, and sometimes there are several competing reasonable estimators or there are no simple estimator at all.
For example, mean and variance of Poisson distribution equal the lambda parameter. However in a Poisson sample, mean and variance are likely to be different - although similar. Should we estimate lambda as sample mean or as sample variance? Finding the ML estimator can be a good choice.
A: The keyword is "estimation". You can not calculate the exact parameter of a distribution from finite observations, only an estimate of it. Under certain regularity conditions, MLE has good properties (efficiency, asymptotic normality, consistency). Therefore, if you don't know what is the "best" estimation of a parameter, you can use MLE to find it.
Intuitively, MLE shows you what to calculate from your observations to estimate the parameter of interest.
