update week37

doc/pub/week37/html/._week37-bs028.html

@@ -293,10 +293,7 @@ mean value of the PDF \( p(x) \) and whose variance is the variance
 of the PDF \( p(x) \) divided by \( m \), the number of values used to compute \( z \).
 
 <p>
-The theorem is satisfied by a large class of PDFs. Note however that for a
-finite \( m \), it is not always possible to find a closed expression for
-\( \tilde{p}(x) \).
-The central limit theorem leads then to the well-known expression for the
+The central limit theorem leads to the well-known expression for the
 standard deviation, given by 
 
 $$
@@ -315,7 +312,17 @@ $$
 
 <p>
 In many cases however the above estimate for the standard deviation,
-in particular if correlations are strong, may be too simplistic.
+in particular if correlations are strong, may be too simplistic. Keep
+in mind that we have assumed that the variables \( x \) are independent
+and identically distributed. This is obviously not always the
+case. For example, the random numbers (or better pseudorandom numbers)
+we generate in various calculations do always exhibit some
+correlations.
+
+<p>
+The theorem is satisfied by a large class of PDFs. Note however that for a
+finite \( m \), it is not always possible to find a closed form /analytic expression for
+\( \tilde{p}(x) \).
 
 <p>
 <p>

doc/pub/week37/html/._week37-bs029.html

@@ -286,7 +286,7 @@ MathJax.Hub.Config({
 <h2 id="confidence-intervals" class="anchor">Confidence Intervals </h2>
 
 <p>
-Confidence intervals are used in statistics is a type of estimate
+Confidence intervals are used in statistics and represent a type of estimate
 computed from the observed data. This gives a range of values for an
 unknown parameter such as the parameters \( \boldsymbol{\beta} \) from linear regression.
 

doc/pub/week37/html/week37-reveal.html

@@ -1031,10 +1031,7 @@ mean value of the PDF \( p(x) \) and whose variance is the variance
 of the PDF \( p(x) \) divided by \( m \), the number of values used to compute \( z \).
 
 <p>
-The theorem is satisfied by a large class of PDFs. Note however that for a
-finite \( m \), it is not always possible to find a closed expression for
-\( \tilde{p}(x) \).
-The central limit theorem leads then to the well-known expression for the
+The central limit theorem leads to the well-known expression for the
 standard deviation, given by
 
 <p>&nbsp;<br>
@@ -1057,7 +1054,17 @@ $$
 
 <p>
 In many cases however the above estimate for the standard deviation,
-in particular if correlations are strong, may be too simplistic.
+in particular if correlations are strong, may be too simplistic. Keep
+in mind that we have assumed that the variables \( x \) are independent
+and identically distributed. This is obviously not always the
+case. For example, the random numbers (or better pseudorandom numbers)
+we generate in various calculations do always exhibit some
+correlations.
+
+<p>
+The theorem is satisfied by a large class of PDFs. Note however that for a
+finite \( m \), it is not always possible to find a closed form /analytic expression for
+\( \tilde{p}(x) \).
 </section>
 
 
@@ -1065,7 +1072,7 @@ in particular if correlations are strong, may be too simplistic.
 <h2 id="confidence-intervals">Confidence Intervals </h2>
 
 <p>
-Confidence intervals are used in statistics is a type of estimate
+Confidence intervals are used in statistics and represent a type of estimate
 computed from the observed data. This gives a range of values for an
 unknown parameter such as the parameters \( \boldsymbol{\beta} \) from linear regression.
 

doc/pub/week37/html/week37-solarized.html

@@ -1057,10 +1057,7 @@ mean value of the PDF \( p(x) \) and whose variance is the variance
 of the PDF \( p(x) \) divided by \( m \), the number of values used to compute \( z \).
 
 <p>
-The theorem is satisfied by a large class of PDFs. Note however that for a
-finite \( m \), it is not always possible to find a closed expression for
-\( \tilde{p}(x) \).
-The central limit theorem leads then to the well-known expression for the
+The central limit theorem leads to the well-known expression for the
 standard deviation, given by 
 
 $$
@@ -1079,7 +1076,17 @@ $$
 
 <p>
 In many cases however the above estimate for the standard deviation,
-in particular if correlations are strong, may be too simplistic.
+in particular if correlations are strong, may be too simplistic. Keep
+in mind that we have assumed that the variables \( x \) are independent
+and identically distributed. This is obviously not always the
+case. For example, the random numbers (or better pseudorandom numbers)
+we generate in various calculations do always exhibit some
+correlations.
+
+<p>
+The theorem is satisfied by a large class of PDFs. Note however that for a
+finite \( m \), it is not always possible to find a closed form /analytic expression for
+\( \tilde{p}(x) \).
 
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
@@ -1087,7 +1094,7 @@ in particular if correlations are strong, may be too simplistic.
 <h2 id="confidence-intervals">Confidence Intervals </h2>
 
 <p>
-Confidence intervals are used in statistics is a type of estimate
+Confidence intervals are used in statistics and represent a type of estimate
 computed from the observed data. This gives a range of values for an
 unknown parameter such as the parameters \( \boldsymbol{\beta} \) from linear regression.
 

doc/pub/week37/html/week37.html

@@ -1062,10 +1062,7 @@ mean value of the PDF \( p(x) \) and whose variance is the variance
 of the PDF \( p(x) \) divided by \( m \), the number of values used to compute \( z \).
 
 <p>
-The theorem is satisfied by a large class of PDFs. Note however that for a
-finite \( m \), it is not always possible to find a closed expression for
-\( \tilde{p}(x) \).
-The central limit theorem leads then to the well-known expression for the
+The central limit theorem leads to the well-known expression for the
 standard deviation, given by 
 
 $$
@@ -1084,7 +1081,17 @@ $$
 
 <p>
 In many cases however the above estimate for the standard deviation,
-in particular if correlations are strong, may be too simplistic.
+in particular if correlations are strong, may be too simplistic. Keep
+in mind that we have assumed that the variables \( x \) are independent
+and identically distributed. This is obviously not always the
+case. For example, the random numbers (or better pseudorandom numbers)
+we generate in various calculations do always exhibit some
+correlations.
+
+<p>
+The theorem is satisfied by a large class of PDFs. Note however that for a
+finite \( m \), it is not always possible to find a closed form /analytic expression for
+\( \tilde{p}(x) \).
 
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
@@ -1092,7 +1099,7 @@ in particular if correlations are strong, may be too simplistic.
 <h2 id="confidence-intervals">Confidence Intervals </h2>
 
 <p>
-Confidence intervals are used in statistics is a type of estimate
+Confidence intervals are used in statistics and represent a type of estimate
 computed from the observed data. This gives a range of values for an
 unknown parameter such as the parameters \( \boldsymbol{\beta} \) from linear regression.
 

doc/pub/week37/ipynb/week37.ipynb

@@ -1065,10 +1065,7 @@
     "mean value of the PDF $p(x)$ and whose variance is the variance\n",
     "of the PDF $p(x)$ divided by $m$, the number of values used to compute $z$.\n",
     "\n",
-    "The theorem is satisfied by a large class of PDFs. Note however that for a\n",
-    "finite $m$, it is not always possible to find a closed expression for\n",
-    "$\\tilde{p}(x)$.\n",
-    "The central limit theorem leads then to the well-known expression for the\n",
+    "The central limit theorem leads to the well-known expression for the\n",
     "standard deviation, given by"
    ]
   },
@@ -1106,11 +1103,23 @@
    "metadata": {},
    "source": [
     "In many cases however the above estimate for the standard deviation,\n",
-    "in particular if correlations are strong, may be too simplistic.\n",
+    "in particular if correlations are strong, may be too simplistic. Keep\n",
+    "in mind that we have assumed that the variables $x$ are independent\n",
+    "and identically distributed. This is obviously not always the\n",
+    "case. For example, the random numbers (or better pseudorandom numbers)\n",
+    "we generate in various calculations do always exhibit some\n",
+    "correlations.\n",
+    "\n",
+    "\n",
+    "\n",
+    "The theorem is satisfied by a large class of PDFs. Note however that for a\n",
+    "finite $m$, it is not always possible to find a closed form /analytic expression for\n",
+    "$\\tilde{p}(x)$.\n",
+    "\n",
     "\n",
     "## Confidence Intervals\n",
     "\n",
-    "Confidence intervals are used in statistics is a type of estimate\n",
+    "Confidence intervals are used in statistics and represent a type of estimate\n",
     "computed from the observed data. This gives a range of values for an\n",
     "unknown parameter such as the parameters $\\boldsymbol{\\beta}$ from linear regression.\n",
     "\n",

doc/src/week37/week37.do.txt

@@ -747,10 +747,7 @@ is a normal distribution whose mean is the
 mean value of the PDF $p(x)$ and whose variance is the variance
 of the PDF $p(x)$ divided by $m$, the number of values used to compute $z$.
 
-The theorem is satisfied by a large class of PDFs. Note however that for a
-finite $m$, it is not always possible to find a closed expression for
-$\tilde{p}(x)$.
-The central limit theorem leads then to the well-known expression for the
+The central limit theorem leads to the well-known expression for the
 standard deviation, given by 
 
 !bt
@@ -771,12 +768,24 @@ the familiar expression in statistics
 !et
 
 In many cases however the above estimate for the standard deviation,
-in particular if correlations are strong, may be too simplistic.
+in particular if correlations are strong, may be too simplistic. Keep
+in mind that we have assumed that the variables $x$ are independent
+and identically distributed. This is obviously not always the
+case. For example, the random numbers (or better pseudorandom numbers)
+we generate in various calculations do always exhibit some
+correlations.
+
+
+
+The theorem is satisfied by a large class of PDFs. Note however that for a
+finite $m$, it is not always possible to find a closed form /analytic expression for
+$\tilde{p}(x)$.
+
 
 !split
 ===== Confidence Intervals =====
 
-Confidence intervals are used in statistics is a type of estimate
+Confidence intervals are used in statistics and represent a type of estimate
 computed from the observed data. This gives a range of values for an
 unknown parameter such as the parameters $\bm{\beta}$ from linear regression.