corrected some pages

Several small mistakes/errors were corrected in several proofs/definitions.

Author: JoramSoch

P/cf-fct.md

@@ -21,7 +21,7 @@ sources:
     pages: "retrieved on 2021-09-22"
     url: "https://www.statlect.com/fundamentals-of-probability/functions-of-random-vectors"
 
-proof_id: "P260"
+proof_id: "P259"
 shortcut: "cf-fct"
 username: "JoramSoch"
 ---

P/mean-lotus.md

@@ -20,6 +20,12 @@ sources:
     in: "Wikipedia, the free encyclopedia"
     pages: "retrieved on 2020-07-22"
     url: "https://en.wikipedia.org/wiki/Law_of_the_unconscious_statistician#Proof"
+  - authors: "Taboga, Marco"
+    year: 2017
+    title: "Transformation theorem"
+    in: "Lectures on probability and mathematical statistics"
+    pages: "retrieved on 2021-09-22"
+    url: "https://www.statlect.com/glossary/transformation-theorem"
 
 proof_id: "P138"
 shortcut: "mean-lotus"

P/mgf-fct.md

@@ -21,7 +21,7 @@ sources:
     pages: "retrieved on 2021-09-22"
     url: "https://www.statlect.com/fundamentals-of-probability/functions-of-random-vectors"
 
-proof_id: "P259"
+proof_id: "P260"
 shortcut: "mgf-fct"
 username: "JoramSoch"
 ---

P/pdf-invfct.md

@@ -52,7 +52,7 @@ sources:
     url: "https://www.youtube.com/watch?v=wUF-lyyWpUc"
 
 proof_id: "P254"
-shortcut: "pmf-invfct"
+shortcut: "pdf-invfct"
 username: "JoramSoch"
 ---
 

P/pmf-invfct.md

@@ -27,7 +27,7 @@ username: "JoramSoch"
 ---
 
 
-**Theorem:** Let $X$ be an $n \times 1$ [random vector](/D/rvec) and let $g: \; \mathbb{R}^n \rightarrow \mathbb{R}^n$ be an invertible function on the support of $X$. Then, the [probability mass function](/D/pmf) of $Y = g(X)$ is given by
+**Theorem:** Let $X$ be an $n \times 1$ [random vector](/D/rvec) with possible outcomes $\mathcal{X}$ and let $g: \; \mathbb{R}^n \rightarrow \mathbb{R}^n$ be an invertible function on the support of $X$. Then, the [probability mass function](/D/pmf) of $Y = g(X)$ is given by
 
 $$ \label{eq:pmf-invfct}
 f_Y(y) = \left\{
@@ -41,7 +41,7 @@ $$
 where $g^{-1}(y)$ is the inverse function of $g(x)$ and $\mathcal{Y}$ is the set of possible outcomes of $Y$:
 
 $$ \label{eq:Y-range}
-\mathcal{Y} = \left\lbrace y = g(x): x \in \mathbb{R}^n \right\rbrace \; .
+\mathcal{Y} = \left\lbrace y = g(x): x \in \mathcal{X} \right\rbrace \; .
 $$