-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 $$ @@ -315,7 +312,17 @@ $$
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 +\( \tilde{p}(x) \).
diff --git a/doc/pub/week37/html/._week37-bs029.html b/doc/pub/week37/html/._week37-bs029.html index e12afed29..6329b76b7 100644 --- a/doc/pub/week37/html/._week37-bs029.html +++ b/doc/pub/week37/html/._week37-bs029.html @@ -286,7 +286,7 @@ MathJax.Hub.Config({
-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. diff --git a/doc/pub/week37/html/week37-reveal.html b/doc/pub/week37/html/week37-reveal.html index 9e19e7ed2..f48a11faa 100644 --- a/doc/pub/week37/html/week37-reveal.html +++ b/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 \( z \).
-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
@@ -1057,7 +1054,17 @@ $$
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 +\( \tilde{p}(x) \). @@ -1065,7 +1072,7 @@ in particular if correlations are strong, may be too simplistic.
-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. diff --git a/doc/pub/week37/html/week37-solarized.html b/doc/pub/week37/html/week37-solarized.html index e551dae68..2178da4cf 100644 --- a/doc/pub/week37/html/week37-solarized.html +++ b/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 \( z \).
-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 $$ @@ -1079,7 +1076,17 @@ $$
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 +\( \tilde{p}(x) \).
@@ -1087,7 +1094,7 @@ in particular if correlations are strong, may be too simplistic.
-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. diff --git a/doc/pub/week37/html/week37.html b/doc/pub/week37/html/week37.html index 0ce52ca65..9c67887f6 100644 --- a/doc/pub/week37/html/week37.html +++ b/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 \( z \).
-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 $$ @@ -1084,7 +1081,17 @@ $$
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 +\( \tilde{p}(x) \).
@@ -1092,7 +1099,7 @@ in particular if correlations are strong, may be too simplistic.
-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. diff --git a/doc/pub/week37/ipynb/ipynb-week37-src.tar.gz b/doc/pub/week37/ipynb/ipynb-week37-src.tar.gz index d3be97922..5f1c144d1 100644 Binary files a/doc/pub/week37/ipynb/ipynb-week37-src.tar.gz and b/doc/pub/week37/ipynb/ipynb-week37-src.tar.gz differ diff --git a/doc/pub/week37/ipynb/week37.ipynb b/doc/pub/week37/ipynb/week37.ipynb index 8dc9800ca..65168ab5c 100644 --- a/doc/pub/week37/ipynb/week37.ipynb +++ b/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", diff --git a/doc/src/week37/week37.do.txt b/doc/src/week37/week37.do.txt index 5ec62aa23..ab8b7e895 100644 --- a/doc/src/week37/week37.do.txt +++ b/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.