3.13 Obtain the statistical maximum likelihood estimates (MLE) of probability distributions

This example uses the normal distribution and Poisson as examples. The maximum likelihood estimates of the population parameters is found by solving equation(s) using the standard method of taking logs and differentiation to solve for the maximum. Mathematica and Matlab version will be added at later time.

3.13.1 Maple

restart;   with(Statistics): 
X := RandomVariable(Normal(mu,sigma)); 
lik:=product(PDF(X,x[i]),i=1..n); 
lik:=expand(ln(lik)) assuming positive; 
eq1:=diff(lik,mu)=0; 
eq2:=diff(lik,sigma)=0; 
solve({eq1,eq2},{mu,sigma}); 
allvalues( %[2]);