corrected some pages

Several small corrections were done to several proofs and definitions.

Author: JoramSoch

P/bvn-mi.md

@@ -58,7 +58,7 @@ $$ \label{eq:mvn-marg}
 X_1 \sim \mathcal{N}\left( \mu_1, \Sigma_{11} \right) \; ,
 $$
 
-such that the [marginals](/D/marg) of the [bivariate normal distribution](/D/bvn) are [univariate normal distribution](/D/norm):
+such that the [marginals](/D/dist-marg) of the [bivariate normal distribution](/D/bvn) are [univariate normal distributions](/D/norm):
 
 $$ \label{eq:bvn-marg}
 \left[ \begin{matrix} X \\ Y \end{matrix} \right] \sim

P/corr-range.md

@@ -59,7 +59,7 @@ $$
 Using the [relationship between covariance and correlation](/P/cov-corr), we have:
 
 $$ \label{eq:var-XY-s2}
-\mathrm{Var}\left( \frac{X}{\sigma_X} \pm \frac{Y}{\sigma_Y} \right) = 1 + 1 + \pm 2 \, \mathrm{Corr}(X,Y) \; .
+\mathrm{Var}\left( \frac{X}{\sigma_X} \pm \frac{Y}{\sigma_Y} \right) = 1 + 1 \pm 2 \, \mathrm{Corr}(X,Y) \; .
 $$
 
 Thus, the combination of \eqref{eq:var-XY-0} with \eqref{eq:var-XY-s2} yields

P/exp-var.md

@@ -77,8 +77,8 @@ Using the following anti-derivative
 
 $$ \label{eq:second-moment-s2}
 \begin{split}
-\int x^2 \cdot \exp(-\lambda x) \, \mathrm{d}x &= \left[ - \frac{1}{\lambda} x^2 \cdot \mathrm{exp}(-\lambda x) \right]_{0}^{+\infty} - \int 2x \left( - \frac{1}{\lambda} x \cdot \mathrm{exp}(-\lambda x) \right) \mathrm{d}x \\
-&= \left[ - \frac{1}{\lambda} x^2 \cdot \mathrm{exp}(-\lambda x) \right]_{0}^{+\infty} - \left( \left[ \frac{1}{\lambda^2} 2x \cdot \mathrm{exp}(-\lambda x) \right]_{0}^{+\infty} - \int 2 \left( \frac{1}{\lambda^2} \cdot \mathrm{exp}(-\lambda x) \right) \mathrm{d}x \right) \\
+\int_{0}^{+\infty} x^2 \cdot \exp(-\lambda x) \, \mathrm{d}x &= \left[ - \frac{1}{\lambda} x^2 \cdot \mathrm{exp}(-\lambda x) \right]_{0}^{+\infty} - \int_{0}^{+\infty} 2x \left( - \frac{1}{\lambda} x \cdot \mathrm{exp}(-\lambda x) \right) \mathrm{d}x \\
+&= \left[ - \frac{1}{\lambda} x^2 \cdot \mathrm{exp}(-\lambda x) \right]_{0}^{+\infty} - \left( \left[ \frac{1}{\lambda^2} 2x \cdot \mathrm{exp}(-\lambda x) \right]_{0}^{+\infty} - \int_{0}^{+\infty} 2 \left( \frac{1}{\lambda^2} \cdot \mathrm{exp}(-\lambda x) \right) \mathrm{d}x \right) \\
 &= \left[ - \frac{x^2}{\lambda} \cdot \mathrm{exp}(-\lambda x) \right]_{0}^{+\infty} - \left( \left[ \frac{2x}{\lambda^2} \cdot \mathrm{exp}(-\lambda x) \right]_{0}^{+\infty} - \left[ - \frac{2}{\lambda^3} \cdot \mathrm{exp}(-\lambda x) \right]_{0}^{+\infty} \right) \\
 &= \left[ \left( - \frac{x^2}{\lambda} - \frac{2x}{\lambda^2} - \frac{2}{\lambda^3} \right) \exp(-\lambda x) \right]_{0}^{+\infty} \; ,
 \end{split}

P/glm-ols.md

@@ -14,6 +14,12 @@ topic: "General linear model"
 theorem: "Ordinary least squares"
 
 sources:
+  - authors: "Wikipedia"
+    year: 2025
+    title: "Ordinary least squares"
+    in: "Wikipedia, the free encyclopedia"
+    pages: "retrieved on 2025-12-19"
+    url: "https://en.wikipedia.org/wiki/Ordinary_least_squares#Assumptions"
 
 proof_id: "P106"
 shortcut: "glm-ols"

P/mvn-mi.md

@@ -58,7 +58,7 @@ $$ \label{eq:mvn-marg}
 X_1 \sim \mathcal{N}\left( \mu_1, \Sigma_{11} \right) \; ,
 $$
 
-such that the [marginals](/D/marg) of $X$ and $Y$ are:
+such that the [marginals](/D/dist-marg) of $X$ and $Y$ are:
 
 $$ \label{eq:X-Y-marg}
 X \sim \mathcal{N}\left( \mu_1, \Sigma_1 \right)

P/norm-var.md

@@ -20,6 +20,12 @@ sources:
     in: "StackExchange Mathematics"
     pages: "retrieved on 2020-01-09"
     url: "https://math.stackexchange.com/questions/518281/how-to-derive-the-mean-and-variance-of-a-gaussian-random-variable"
+  - authors: "Wikipedia"
+    year: 2025
+    title: "Particular values of the gamma function"
+    in: "Wikipedia, the free encyclopedia"
+    pages: "retrieved on 2025-12-19"
+    url: "https://en.wikipedia.org/wiki/Particular_values_of_the_gamma_function#Integers_and_half-integers"
 
 proof_id: "P18"
 shortcut: "norm-var"

P/rsq-der.md

@@ -61,7 +61,6 @@ where $X$ is the $n \times p$ design matrix and $\hat{\beta}$ are the [ordinary
 
 **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
 
 $$ \label{eq:ESS}
@@ -80,9 +79,12 @@ $$ \label{eq:R2-s2}
 R^2 = \frac{\mathrm{TSS}-\mathrm{RSS}}{\mathrm{TSS}} = 1 - \frac{\mathrm{RSS}}{\mathrm{TSS}} \; ,
 $$
 
-[because](/P/mlr-pss) $\mathrm{TSS} = \mathrm{ESS} + \mathrm{RSS}$.
+[because it holds that](/P/mlr-pss)
+
+$$ \label{eq:TSS-ESS-RSS}
+\mathrm{TSS} = \mathrm{ESS} + \mathrm{RSS} \; .
+$$
 
-<br>
 2) Using \eqref{eq:SS}, the coefficient of determination can be also written as:
 
 $$ \label{eq:R2'}
@@ -95,7 +97,4 @@ $$ \label{eq:R2-adj'}
 R^2_{\mathrm{adj}} = 1 - \frac{\frac{1}{n-p} \sum_{i=1}^{n} (y_i - \hat{y}_i)^2}{\frac{1}{n-1} \sum_{i=1}^{n} (y_i - \bar{y})^2} = 1 - \frac{\mathrm{RSS}/\mathrm{df}_r}{\mathrm{TSS}/\mathrm{df}_t}
 $$
 
-where $\mathrm{df}_r = n-p$ and $\mathrm{df}_t = n-1$ are the residual and total [degrees of freedom](/D/dof).
-
-<br>
-This gives the adjusted $R^2$ which adjusts $R^2$ for the number of explanatory variables.
+where $\mathrm{df}_r = n-p$ and $\mathrm{df}_t = n-1$ are the residual and total [degrees of freedom](/D/dof). This gives the adjusted $R^2$ which adjusts $R^2$ for the [number of explanatory variables](/D/mlr).

P/slr-rescorr.md

@@ -27,15 +27,15 @@ username: "JoramSoch"
 ---
 
 
-**Theorem:** In [simple linear regression](/D/slr), the [residuals](/D/rss) and the [covariate](/D/slr) are [uncorrelated](/D/corr) when estimated using [ordinary least squares](/P/slr-ols).
+**Theorem:** In [simple linear regression](/D/slr), the [residuals](/D/rss) and the [covariate](/D/slr) are [uncorrelated](/D/corr) when [estimated using ordinary least squares](/P/slr-ols).
 
 **Proof:** The residuals are defined as the estimated [error terms](/D/slr)
 
 $$ \label{eq:slr-res}
 \hat{\varepsilon}_i = y_i - \hat{\beta}_0 - \hat{\beta}_1 x_i
 $$
 
-where $\hat{\beta}_0$ and $\hat{\beta}_1$ are parameter estimates obtained using [ordinary least squares](/P/slr-ols):
+where $\hat{\beta}_0$ and $\hat{\beta}_1$ are parameter estimates [obtained using ordinary least squares](/P/slr-ols):
 
 $$ \label{eq:slr-ols}
 \hat{\beta}_0 = \bar{y} - \hat{\beta}_1 \bar{x} \quad \text{and} \quad \hat{\beta}_1 = \frac{s_{xy}}{s_x^2} \; .
@@ -60,4 +60,4 @@ $$ \label{eq:slr-rescorr}
 \end{split}
 $$
 
-Because an inner product of zero also implies zero [correlation](/D/corr), this demonstrates that [residuals](/D/rss) and [covariate](/D/slr) values are uncorrelated under [ordinary least squares](/P/slr-ols).
+Because an inner product of zero also implies zero [correlation](/D/corr), this demonstrates that [residuals](/D/rss) and [covariate](/D/slr) values are uncorrelated [under ordinary least squares](/P/slr-ols).