For example suppose i am given a data set X in R^n which is basically a bunch of data points and I wanted to determine what the distribution mean is. I would then consider which is the most likely value based on what I know. If I assume the data comes from the normal distribution N(mu,sigma^2) with mu as the mean and sigma^2 as the variance then we have f(X|mu,sigma^2) =prod_i^n 1/sqrt(2pi sigma ^2)e^(-1/(2sigma^2)(x_i-mu)^2).
If mu is not known then I would try to estimate it by way of maximum likelihood or using the equation I would state
l(mu|X,sigma^2)=prod_i^n 1/sqrt(2pi sigma ^2)e^(-1/(2sigma^2)(x_i-mu)^2)
Here the equation is the same but the paramter of interest is mu. To solve we take the derivative, set it equal to 0 and solve for mu so we have.
(partial)/(partialmu) prod_i^n 1/sqrt(2pi sigma ^2)e^(-1/(2sigma^2)(x_i-mu)^2)
However before doing so I see that I can apply the natural log before finding the derivative to solve for x and simplify the equation thus ...
ln(l(mu|X,sigma^2))= sum_i^n ln(1/sqrt(2pi sigma ^2)) -1/(2sigma^2)(x_i-mu)^2
(partial)/(partialmu) sum_i^n ln(1/sqrt(2pi sigma ^2)) -1/(2sigma^2)(x_i-mu)^2
= 1/(sigma^2)sum_i^n(x_i-mu) =0
= 1/(sigma^2)sum_i^nx_i = 1/(sigma^2)sum_i^nmu
= sum_i^nx_i = n*mu
= 1/nsum_i^nx_i = mu
so an approximation of mu would be the average of the data or barx = 1/nsum_i^nx_i.
Using MLE we can also find out what the estimated standard deviation is.
