corrected some pages

JoramSoch · JoramSoch · commit 1deb9cc263d0 · 2022-12-23T17:18:11.000+01:00
Several small mistakes/errors were corrected in several proofs/definitions.
diff --git a/P/bin-pmf.md b/P/bin-pmf.md
@@ -34,7 +34,7 @@ f_X(x) = {n \choose x} \, p^x \, (1-p)^{n-x} \; .
 $$
 
 
-**Proof:** A binomial variable [is defined as](/D/bin) the number of successes observed in $n$ [independent](/D/ind) trials, where each trial has [two possible outcomes](/D/bern) (success/failure) and the [probability](/D/prob) of success and failure are identical across trials ($p$, $q = 1-p$).
+**Proof:** A [binomial variable](/D/bin) is defined as the number of successes observed in $n$ [independent](/D/ind) trials, where each trial has [two possible outcomes](/D/bern) (success/failure) and the [probability](/D/prob) of success and failure are identical across trials ($p$, $q = 1-p$).
 
 If one has obtained $x$ successes in $n$ trials, one has also obtained $(n-x)$ failures. The probability of a particular series of $x$ successes and $(n-x)$ failures, when order does matter, is
 
diff --git a/P/cuni-qf.md b/P/cuni-qf.md
@@ -51,7 +51,7 @@ F_X(x) = \left\{
 \right.
 $$
 
-The quantile function $Q_X(p)$ [is defined as](/D/qf) the smallest $x$, such that $F_X(x) = p$:
+The [quantile function](/D/qf) $Q_X(p)$ is defined as the smallest $x$, such that $F_X(x) = p$:
 
 $$ \label{eq:qf}
 Q_X(p) = \min \left\lbrace x \in \mathbb{R} \, \vert \, F_X(x) = p \right\rbrace \; .
diff --git a/P/duni-qf.md b/P/duni-qf.md
@@ -53,7 +53,7 @@ F_X(x) = \left\{
 \right.
 $$
 
-The quantile function $Q_X(p)$ [is defined as](/D/qf) the smallest $x$, such that $F_X(x) = p$:
+The [quantile function](/D/qf) $Q_X(p)$ is defined as the smallest $x$, such that $F_X(x) = p$:
 
 $$ \label{eq:qf}
 Q_X(p) = \min \left\lbrace x \in \mathbb{R} \, \vert \, F_X(x) = p \right\rbrace \; .
diff --git a/P/exp-qf.md b/P/exp-qf.md
@@ -50,7 +50,7 @@ F_X(x) = \left\{
 \right.
 $$
 
-The quantile function $Q_X(p)$ [is defined as](/D/qf) the smallest $x$, such that $F_X(x) = p$:
+The [quantile function](/D/qf) $Q_X(p)$ is defined as the smallest $x$, such that $F_X(x) = p$:
 
 $$ \label{eq:qf}
 Q_X(p) = \min \left\lbrace x \in \mathbb{R} \, \vert \, F_X(x) = p \right\rbrace \; .
diff --git a/P/gam-qf.md b/P/gam-qf.md
@@ -58,7 +58,7 @@ F_X(x) = \left\{
 \right.
 $$
 
-The quantile function $Q_X(p)$ [is defined as](/D/qf) the smallest $x$, such that $F_X(x) = p$:
+The [quantile function](/D/qf) $Q_X(p)$ is defined as the smallest $x$, such that $F_X(x) = p$:
 
 $$ \label{eq:qf}
 Q_X(p) = \min \left\lbrace x \in \mathbb{R} \, \vert \, F_X(x) = p \right\rbrace \; .
diff --git a/P/mlr-f.md b/P/mlr-f.md
@@ -121,7 +121,7 @@ $$ \label{eq:mlr-rss-dist}
 \frac{\hat{\varepsilon}^\mathrm{T} \hat{\varepsilon}}{\sigma^2} = \tau \, \hat{\varepsilon}^\mathrm{T} \hat{\varepsilon} \sim \chi^2(n-p) \; .
 $$
 
-The [chi-squared distribution is related to the gamma distribution](/P/gam-chi2) in the following way:
+The [chi-squared distribution is related to the gamma distribution](/P/chi2-gam) in the following way:
 
 $$ \label{eq:gam-chi2}
 X \sim \chi^2(k) \quad \Rightarrow \quad cX \sim \mathrm{Gam}\left( \frac{k}{2}, \frac{1}{2c} \right) \; .
diff --git a/P/mlr-olsdist.md b/P/mlr-olsdist.md
@@ -11,7 +11,7 @@ title: "Distributions of estimated parameters, fitted signal and residuals in mu
 chapter: "Statistical Models"
 section: "Univariate normal data"
 topic: "Multiple linear regression"
-theorem: "Distribution of OLS estimates, fitted signal and residuals"
+theorem: "Distribution of OLS estimates, signal and residuals"
 
 sources:
   - authors: "Koch, Karl-Rudolf"
diff --git a/P/mlr-rssdist.md b/P/mlr-rssdist.md
@@ -85,7 +85,7 @@ $$ \label{eq:mvn-cochran}
 x \sim \mathcal{N}(\mu, \sigma^2 I_n) \quad \Rightarrow \quad y = x^\mathrm{T} A x /\sigma^2 \sim \chi^2\left( \mathrm{tr}(A), \mu^\mathrm{T} A \mu \right) \; .
 $$
 
-First, we formulate the residuals in terms of transformed measurements $\tilde{y}$:
+First, we [formulate the residuals](/P/mlr-mat) in terms of transformed measurements $\tilde{y}$:
 
 $$ \label{eq:rss-y-s1}
 \begin{array}{rlcl}
diff --git a/P/mlr-wlsdist.md b/P/mlr-wlsdist.md
@@ -11,7 +11,7 @@ title: "Distributions of estimated parameters, fitted signal and residuals in mu
 chapter: "Statistical Models"
 section: "Univariate normal data"
 topic: "Multiple linear regression"
-theorem: "Distribution of estimated parameters, signal and residuals"
+theorem: "Distribution of WLS estimates, signal and residuals"
 
 sources:
   - authors: "Koch, Karl-Rudolf"
@@ -116,7 +116,7 @@ $$
 3) The [residuals of the linear regression model](/P/mlr-mat) are given by
 
 $$ \label{eq:e-est}
-\hat{\varepsilon} = y - X \hat{\beta} = \left( I_n - X (X^\mathrm{T} V^{-1} X)^{-1} X^\mathrm{T} \right) y = \left( I_n - P \right) y
+\hat{\varepsilon} = y - X \hat{\beta} = \left( I_n - X (X^\mathrm{T} V^{-1} X)^{-1} X^\mathrm{T} V^{-1} \right) y = \left( I_n - P \right) y
 $$
 
 and thus, by applying \eqref{eq:mvn-ltt} to \eqref{eq:e-est}, they are distributed as
@@ -126,9 +126,9 @@ $$ \label{eq:e-est-dist}
 \hat{\varepsilon} &\sim \mathcal{N}\left( \left[ I_n - X (X^\mathrm{T} V^{-1} X)^{-1} X^\mathrm{T} V^{-1} \right] X \beta, \, \sigma^2 \left[ I_n - P \right] V \left[ I_n - P \right]^\mathrm{T} \right) \\
 &\sim \mathcal{N}\left( X \beta - X \beta, \, \sigma^2 \left[ V - V P^\mathrm{T} - P V + P V P^\mathrm{T} \right] \right) \\
 &\sim \mathcal{N}\left( 0, \, \sigma^2 \left[ V - V V^{-1} X (X^\mathrm{T} V^{-1} X)^{-1} X^\mathrm{T} - X (X^\mathrm{T} V^{-1} X)^{-1} X^\mathrm{T} V^{-1} V + P V P^\mathrm{T} \right] \right) \\
-&\sim \mathcal{N}\left( 0, \, \sigma^2 \left[ V - 2 P + X (X^\mathrm{T} V^{-1} X)^{-1} X^\mathrm{T} V^{-1} V V^{-1} X (X^\mathrm{T} V^{-1} X)^{-1} X^\mathrm{T} \right] \right) \\
-&\sim \mathcal{N}\left( 0, \, \sigma^2 \left[ V - 2 P + P \right] \right) \\
-&\sim \mathcal{N}\left( 0, \, \sigma^2 \left[ V - X (X^\mathrm{T} V^{-1} X)^{-1} X^\mathrm{T} \right] \right) \\
+&\sim \mathcal{N}\left( 0, \, \sigma^2 \left[ V - 2 P V + X (X^\mathrm{T} V^{-1} X)^{-1} X^\mathrm{T} V^{-1} V V^{-1} X (X^\mathrm{T} V^{-1} X)^{-1} X^\mathrm{T} \right] \right) \\
+&\sim \mathcal{N}\left( 0, \, \sigma^2 \left[ V - 2 P V + P V \right] \right) \\
+&\sim \mathcal{N}\left( 0, \, \sigma^2 \left[ V - P V \right] \right) \\
 &\sim \mathcal{N}\left( 0, \, \sigma^2 \left[ I_n - P \right] V \right) \; .
 \end{split}
 $$
diff --git a/P/mult-pmf.md b/P/mult-pmf.md
@@ -34,7 +34,7 @@ f_X(x) = {n \choose {x_1, \ldots, x_k}} \, \prod_{i=1}^k {p_i}^{x_i} \; .
 $$
 
 
-**Proof:** A multinomial variable [is defined as](/D/mult) a vector of the numbers of observations belonging to $k$ distinct categories in $n$ [independent](/D/ind) trials, where each trial has [$k$ possible outcomes](/D/cat) and the category [probabilities](/D/prob) are identical across trials.
+**Proof:** A [multinomial variable](/D/mult) is defined as a vector of the numbers of observations belonging to $k$ distinct categories in $n$ [independent](/D/ind) trials, where each trial has [$k$ possible outcomes](/D/cat) and the category [probabilities](/D/prob) are identical across trials.
 
 The probability of a particular series of $x_1$ observations for category $1$, $x_2$ observations for category $2$ etc., when order does matter, is
 
diff --git a/P/rsq-der.md b/P/rsq-der.md
@@ -59,7 +59,7 @@ $$
 where $X$ is the $n \times p$ design matrix and $\hat{\beta}$ are the [ordinary least squares](/P/mlr-ols) estimates.
 
 
-**Proof:** The coefficient of determination $R^2$ [is defined as](/D/rsq) the proportion of the variance explained by the independent variables, relative to the total variance in the data.
+**Proof:** The [coefficient of determination](/D/rsq) $R^2$ is defined as the proportion of the variance explained by the independent variables, relative to the total variance in the data.
 
 <br>
 1) If we define the [explained sum of squares](/D/ess) as
diff --git a/P/snr-rsq.md b/P/snr-rsq.md
@@ -48,7 +48,7 @@ $$
 if the predicted signal mean is equal to the actual signal mean.
 
 
-**Proof:** The signal-to-noise ratio (SNR) [is defined as](/D/snr)
+**Proof:** The [signal-to-noise ratio](/D/snr) (SNR) is defined as
 
 $$ \label{eq:SNR}
 \mathrm{SNR} = \frac{\mathrm{Var}(X\hat{\beta})}{\hat{\sigma}^2} = \frac{\mathrm{Var}(\hat{y})}{\hat{\sigma}^2} \; .
