General update of several files with to do list

doc/src/Statistics/Statistics.do.txt

@@ -1,7 +1,11 @@
-TITLE: Data Analysis and Machine Learning: Elements of Probability Theory
+TITLE: Data Analysis and Machine Learning: Elements of Probability Theory and Statistical Data Analysis
 AUTHOR: Morten Hjorth-Jensen {copyright, 1999-present|CC BY-NC} at Department of Physics, University of Oslo & Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University
 DATE: today
 
+!split
+===== Things to add =====
+Add general statistic elements (probability theory mainly), assumed knowledge
+
 
 !split
 ===== Domains and probabilities  =====
@@ -568,6 +572,14 @@ the binomial distribution we can show that
 !et
 !eblock
 
+
+!split
+===== Additions to make =====
+* discuss more sample mean and variance
+* sample covariance and Bessel's theorem on 1/(n-1) versus 1/n
+* add more text to covariance matrix and results of codes
+
+
 !split
 ===== Meet the  covariance! =====
 !bblock
@@ -949,6 +961,49 @@ more practically oriented methods like the blocking technique.
 #add ref here to flybjerg
 !eblock
 
+
+!split 
+===== Code to compute the Covariance matrix and the Covariance =====
+!bc pycod
+# Importing various packages
+from math import exp, sqrt
+from random import random, seed
+import numpy as np
+import matplotlib.pyplot as plt
+
+# Sample covariance, note the factor 1/(n-1)
+def covariance(x, y, n):
+    sum = 0.0
+    mean_x = np.mean(x)
+    mean_y = np.mean(y)
+    for i in range(0, n):
+        sum += (x[(i)]-mean_x)*(y[i]-mean_y)
+    return  sum/(n-1.)
+
+n = 100
+x = np.random.normal(size=n)
+print(np.mean(x))
+y = 4+3*x+np.random.normal(size=n)
+print(np.mean(y))
+z = x**3+np.random.normal(size=n)
+print(np.mean(z))
+covxx = covariance(x,x,n)
+covyy = covariance(y,y,n)
+covzz = covariance(z,z,n)
+covxy = covariance(x,y,n)
+covxz = covariance(x,z,n)
+covyz = covariance(y,z,n)
+print(covxx,covyy, covzz)
+print(covxy,covxz, covyz)
+w = np.vstack((x, y, z))
+#print(w)
+c = np.cov(w)
+print(c)
+#eigen = np.zeros(n)
+Eigvals, Eigvecs = np.linalg.eig(c)
+print(Eigvals)
+!ec
+
 !split
 ======= Random Numbers =======
 !bblock