%David Agrawal
%Economics 671
%Problem Set 10 Due Monday November 5, 2007

format('long','g');
close all; %closes all figures
clc; %clears the screen
clear; %clears the memory

%1.a.
%set the normal seed
rand('seed',0);

%Set Matrix Dim
m=100;
n=1;

%Set Binomial Parameters
%Bernoulli Draws, nbio=1
nbio=1;
p=0.52;

y=binornd(nbio,p,m,n);

%1.b.  Calculations Attached
%MLE for Bernoulli is Phat = Xbar
%note xbar = mean of p = .52
phat=mean(y);
xbar=mean(y);

%1.c.
% Log Likelihood function.  Note must place dot before matrix operators.
%log likelihood function with p = r
L=@(r) -m.*xbar.*log(r)-m.*(1-xbar).*log(1-r);

%Plot L
fplot(L,[0.001 .999])

%find the minimum of -L and calculate numerical phat starting search at .5
phatnum = fminsearch(L,.1)
%solution is .52

%1.d.
%give the first order condition
dLdr = @(r) m.*xbar./r-m.*(1-xbar)./(1-r)
phatnum2 = fsolve(dLdr,.5)
%solution is .52

%note all solutions yield .52, which should be the case.  The FOC,
%minimization problem and the closed form solution should all eqaul .52