corrected some pages

Several small corrections were done to several proofs and definitions.

Author: JoramSoch

D/est-bias.md

@@ -29,6 +29,6 @@ username: "JoramSoch"
 
 **Definition:** Let $\hat{\theta}: \mathcal{Y} \rightarrow \Theta$ be an [estimator](/D/est) of a [parameter](/D/para) $\theta \in \Theta$ from [data](/D/data) $y \in \mathcal{Y}$. Then,
 
-* $\hat{\theta}$ is called an unbiased estimator when its [expected value](/D/mean) is equal to the parameter that it is estimating: $\mathrm{E}_{\hat{\theta}}\left[ \hat{\theta} \right] = \theta$, where the expectation is calculated over all possible samples $y$ leading to values of $\hat{\theta}$.
+* $\hat{\theta}$ is called an unbiased estimator when its [expected value](/D/mean) is equal to the parameter that it is estimating: $\mathrm{E}_{\hat{\theta}}(\hat{\theta}) = \theta$, where the expectation is calculated over all possible samples $y$ leading to values of $\hat{\theta}$.
 
-* $\hat{\theta}$ is called a biased estimator otherwise, i.e. when $\mathrm{E}_{\hat{\theta}}\left[ \hat{\theta} \right] \neq \theta$.
+* $\hat{\theta}$ is called a biased estimator otherwise, i.e. when $\mathrm{E}_{\hat{\theta}}(\hat{\theta}) \neq \theta$.

D/est.md

@@ -14,6 +14,12 @@ topic: "Basic concepts of estimation"
 definition: "Estimator"
 
 sources:
+  - authors: "Ostwald, Dirk"
+    year: 2023
+    title: "Punktschätzung"
+    in: "Wahrscheinlichkeitstheorie und Frequentistische Inferenz"
+    pages: "Einheit (9), Folie 7"
+    url: "https://www.ipsy.ovgu.de/ipsy_media/Methodenlehre+I/Wintersemester+2324/Wahrscheinlichkeitstheorie+und+Frequentistische+Inferenz/9_Punktsch%C3%A4tzung.pdf"
   - authors: "Wikipedia"
     year: 2024
     title: "Estimator"

D/nw.md

@@ -14,6 +14,12 @@ topic: "Normal-Wishart distribution"
 definition: "Definition"
 
 sources:
+  - authors: "Bishop, Christopher M."
+    year: 2006
+    title: "Appendix B. Probability Distributions"
+    in: "Pattern Recognition for Machine Learning"
+    pages: "p. 690, eq. B.53"
+    url: "http://users.isr.ist.utl.pt/~wurmd/Livros/school/Bishop%20-%20Pattern%20Recognition%20And%20Machine%20Learning%20-%20Springer%20%202006.pdf"
 
 def_id: "D175"
 shortcut: "nw"

P/mgf-lincomb.md

@@ -38,7 +38,7 @@ where $a_1, \ldots, a_n$ are $n$ real numbers.
 
 **Proof:** The [moment-generating function of a random variable](/D/mgf) $X_i$ is
 
-$$ \label{eq:mfg-vect}
+$$ \label{eq:mfg}
 M_{X_i}(t) = \mathrm{E} \left( \exp \left[ t X_i \right] \right)
 $$
 

P/mlr-olstr.md

@@ -39,7 +39,7 @@ $$
 2) and, if the two regressors are orthogonal to each other, they simplify to
 
 $$ \label{eq:mlr-ols-tr-orth}
-\hat{\beta}_1 = \frac{x_1^\mathrm{T} y}{x_1^\mathrm{T} x_1} \quad \text{and} \quad \hat{\beta}_2 = \frac{x_2^\mathrm{T} y}{x_2^\mathrm{T} x_2}, \quad \text{if} \quad x_1 \perp x_2 \; .
+\hat{\beta}_1 = \frac{x_1^\mathrm{T} y}{x_1^\mathrm{T} x_1} \quad \text{and} \quad \hat{\beta}_2 = \frac{x_2^\mathrm{T} y}{x_2^\mathrm{T} x_2} \; .
 $$
 
 

P/nw-pdf.md

@@ -14,6 +14,12 @@ topic: "Normal-Wishart distribution"
 theorem: "Probability density function"
 
 sources:
+  - authors: "Bishop, Christopher M."
+    year: 2006
+    title: "Appendix B. Probability Distributions"
+    in: "Pattern Recognition for Machine Learning"
+    pages: "p. 690, eq. B.53"
+    url: "http://users.isr.ist.utl.pt/~wurmd/Livros/school/Bishop%20-%20Pattern%20Recognition%20And%20Machine%20Learning%20-%20Springer%20%202006.pdf"
 
 proof_id: "P323"
 shortcut: "nw-pdf"