% filename: inference_example_2.m

% This script plots the likelihood function for Example 2 in the Introduction to Inference presentation.  In the example, 22 white marbles and 28 reds were drawn at random from a large bag, and the goal was to give plausible values for pi, the proportion of reds in the bag.

% **********

% create a vector of candidate values of the parameter pi, and compute the likelihood L at each of those values.  Note that .^ and .* are element-by-element exponentiation and multiplication respectively.

p = [0:0.01:1];
L = (p.^28).*((1-p).^22);

% plot the likelihood function:

plot(p,L)
xlabel('\pi')
ylabel('Likelihood L(\pi )')

% plot a red horizontal line at 10% of the maximum likelihood

hold on
MLE = max(L);
plot([0,1],0.1*MLE*[1,1],'r')

% determine ("find") the range of values of pi for which the likelihood is at least 10% of its maximum value, and then report them as an interval.

indices_of_plausibles = find(L>=0.1*MLE);
plausibles = p(indices_of_plausibles);
disp(['Range of plausible values is from ',num2str(min(plausibles)),' to ',num2str(max(plausibles))])