Few updates on todays lecture

statistics/code/gaussianbins.m

@@ -1,6 +1,6 @@
-x = randn( 100, 1 );
-bins1 = -4:2:4;
-bins2 = -4:0.5:4;
+x = randn( 100, 1 );   % generate some data
+bins1 = -4:2:4;        % large bins
+bins2 = -4:0.5:4;      % small bins
 subplot( 1, 2, 1 );
 hold on;
 hist( x, bins1 );
@@ -10,6 +10,7 @@ ylabel('Frequeny')
 hold off;
 subplot( 1, 2, 2 );
 hold on;
+% normalize to the rigtht bin size:
 hist( x, bins1, 1.0/(bins1(2)-bins1(1)) );
 hist( x, bins2, 1.0/(bins2(2)-bins2(1)) );
 xlabel('x')

statistics/code/gaussianpdf.m

@@ -1,22 +1,30 @@
 % plot Gaussian pdf:
-dx=0.1
+dx=0.1;
 x = [-4.0:dx:4.0];
 p = exp(-0.5*x.^2)/sqrt(2.0*pi);
 hold on
-plot(x,p, 'linewidth', 10 )
-
-% compute integral between x1 and x2:
-x1=1.0
-x2=2.0
-P = sum(p((x>=x1)&(x<x2)))*dx
-
-% draw random numbers:
-r = randn( 10000, 1 );
-hist(r,x,1.0/dx)
-
-% check P:
-Pr = sum((r>=x1)&(r<x2))/length(r)
-
+plot(x, p, 'linewidth', 10)
+% show area of integral:
+area(x((x>=x1)&(x<=x2)), p((x>=x1)&(x<=x2)), 'FaceColor', 'r' )
 hold off
 
+% compute integral between x1 and x2:
+x1=1.0;
+x2=2.0;
+P = sum(p((x>=x1)&(x<x2)))*dx;
+disp( [ 'The integral between ', num2str(x1, 1), ' and ', num2str(x2, 1), ' is ', num2str(P, 3) ] );
+
+% draw random numbers:
+%r = randn( 10000, 1 );
+%hist(r,x,1.0/dx)
+
+% check P:
+Pr = sum((r>=x1)&(r<x2))/length(r);
+disp( [ 'The probability of getting a number between ', num2str(x1, 1), ' and ', num2str(x2, 1), ' is ', num2str(Pr, 3) ] );
+
+% infinite integral:
+P = sum(p)*dx;
+disp( [ 'The integral between -infinity and +infinity is ', num2str(P, 3) ] );
+disp( [ 'I.e. the probability to get any number is ', num2str(P, 3) ] );
+
 

statistics/code/mymedian.m

@@ -1,13 +1,13 @@
 function m = mymedian( x )
 % returns the median of the vector x
   xs = sort( x );
-  if ( length( xs ) == 0 )
+  if ( length( xs ) == 0 )                 % empty input vector
     m = NaN;
-  elseif ( rem( length( xs ), 2 ) == 0 )
+  elseif ( rem( length( xs ), 2 ) == 0 )   % even number of data values
     index = length( xs )/2;
-    m = (xs( index ) + xs( index+1 ))/2;
-  else
-    index = (length( xs ) + 1)/2;
+    m = (xs( index ) + xs( index+1 ))/2;   % average the two central elements
+  else                                     % odd number of data values
+    index = (length( xs ) + 1)/2;          % take the middle element
     m = xs( index );
   end
 end

statistics/code/quartiles.m

@@ -1,13 +1,14 @@
 function q = quartiles( x )
-  % returns a vector with the first, second, and third quartile of the vector x
+  % returns a vector with the first, second, and third quartile
+  % of the vector x
   xs = sort( x );
-  if ( length( xs ) == 0 )
+  if ( length( xs ) == 0 )                            % no data
     q = [];
-  elseif ( rem( length( xs ), 2 ) == 0 )
+  elseif ( rem( length( xs ), 2 ) == 0 )              % even number of data
     index = length( xs )/2;
     m = (xs( index ) + xs( index+1 ))/2;
     q = [ round( xs(length(xs)/4) ), m, xs(round(3*length(xs)/4)) ];
-  else
+  else                                                % odd number of data
     index = (length( xs ) + 1)/2;
     m = xs( index );
     q = [ round( xs(length(xs)/4) ), m, xs(round(3*length(xs)/4)) ];