Based on Kitonum's code, but using the built-in PDF for the Normal distribution. The advantage is that you can easily adapt the code to another distribution, if you were so inclined, and you don't have to remember the PDF of the distribution, and you avoid possible mis-typing of the distribution.
Side remark: I did not detect any advantage in terms of speed or memory usage
CodeTools:-Usage(# based on Kitonum
plots:-animate(plot, [Statistics:-PDF(Statistics:-RandomVariable(Normal(mu,1)),x), x=-5..6,'thickness'=2,'color'=blue], mu=[seq(-2+0.02*k,k=0..250)])
For fun, here an animation as the "entropy" H varies between 0 and 1.
h := 1/2*ln(2*Pi*exp(1)*sigma^2);
sig := solve(H=h,sigma,UseAssumptions) assuming sigma>0;
pdfN := Statistics:-PDF(Statistics:-RandomVariable(Normal(mu,sigma)),x):
pdfNs := unapply( eval(pdfN,sigma=sig), (mu,H) ):
plots:-animate( plot, [pdfNs(0,H), x=-5..5,'thickness'=2,'color'=blue], H=[seq(i,i=0..1,0.01)] );