function [] = chap_1_prob_7(n)
% 
% Written by:
% -- 
% John L. Weatherwax                2007-07-01
% 
% email: wax@alum.mit.edu
% 
% Please send comments and especially bug reports to the
% above email address.
% 
%-----

%n = 10; 

%
% Part (a):
%

% Verify that our estimate of the probabilities sum to one: 
%
P_sum = 0; 
for ii=1:n, 
  P_sum = P_sum + prob_select_keys_removed( ii, n );
end
P_sum 

% Compute the expectation E[k]: 
%
Exp = 0; 
for ii=1:n, 
  Exp = Exp + ii*prob_select_keys_removed( ii, n );
end
Exp 

% Compute the expectation squared E[k^2]: 
%
Exp2 = 0; 
for ii=1:n, 
  Exp2 = Exp2 + (ii^2)*prob_select_keys_removed( ii, n );
end
Exp2 

% Compute the variance: 
%
Var = Exp2 - Exp^2 


function p = prob_select_keys_removed( k, n )

switch k
  case 1, 
    p = 1/n; 
  case 2,
    p = (1-1/n)*(1/(n-1)); 
  otherwise,
    p = 1; 
    for ii=0:(k-2),
      p = p*( 1-1/(n-ii) );
    end  
    p = p*(1/(n-(k-1))); 
end