corrected some pages

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

Author: JoramSoch

D/beta.md

@@ -27,7 +27,7 @@ username: "JoramSoch"
 ---
 
 
-**Definition**: Let $X$ be a [random variable](/D/rvar). Then, $X$ is said to follow a beta distribution with shape parameters $\alpha$ and $\beta$
+**Definition:** Let $X$ be a [random variable](/D/rvar). Then, $X$ is said to follow a beta distribution with shape parameters $\alpha$ and $\beta$
 
 $$ \label{eq:beta}
 X \sim \mathrm{Bet}(\alpha, \beta) \; ,

D/dir.md

@@ -27,7 +27,7 @@ username: "JoramSoch"
 ---
 
 
-**Definition**: Let $X$ be a [random vector](/D/rvec). Then, $X$ is said to follow a Dirichlet distribution with concentration parameters $\alpha = \left[ \alpha_1, \ldots, \alpha_k \right]$
+**Definition:** Let $X$ be a [random vector](/D/rvec). Then, $X$ is said to follow a Dirichlet distribution with concentration parameters $\alpha = \left[ \alpha_1, \ldots, \alpha_k \right]$
 
 $$ \label{eq:Dir}
 X \sim \mathrm{Dir}(\alpha) \; ,

D/dist-cond.md

@@ -27,4 +27,4 @@ username: "JoramSoch"
 ---
 
 
-**Definition**: Let $X$ and $Y$ be [random variables](/D/rvar) with sets of possible outcomes $\mathcal{X}$ and $\mathcal{Y}$. Then, the conditional distribution of $X$ given that $Y$ is a [probability distribution](/D/dist) that specifies the probability of the event that $X = x$ given that $Y = y$ for each possible combination of $x \in \mathcal{X}$ and $y \in \mathcal{Y}$. The conditional distribution of $X$ can be obtained from the [joint distribution](/D/dist-joint) of $X$ and $Y$ and the [marginal distribution](/D/dist-marg) of $Y$ using the [law of conditional probability](/D/prob-cond).
+**Definition:** Let $X$ and $Y$ be [random variables](/D/rvar) with sets of possible outcomes $\mathcal{X}$ and $\mathcal{Y}$. Then, the conditional distribution of $X$ given that $Y$ is a [probability distribution](/D/dist) that specifies the probability of the event that $X = x$ given that $Y = y$ for each possible combination of $x \in \mathcal{X}$ and $y \in \mathcal{Y}$. The conditional distribution of $X$ can be obtained from the [joint distribution](/D/dist-joint) of $X$ and $Y$ and the [marginal distribution](/D/dist-marg) of $Y$ using the [law of conditional probability](/D/prob-cond).

D/dist-joint.md

@@ -27,7 +27,7 @@ username: "JoramSoch"
 ---
 
 
-**Definition**: Let $X$ and $Y$ be [random variables](/D/rvar) with sets of possible outcomes $\mathcal{X}$ and $\mathcal{Y}$. Then, a joint distribution of $X$ and $Y$ is a [probability distribution](/D/dist) that specifies the probability of the event that $X = x$ and $Y = y$ for each possible combination of $x \in \mathcal{X}$ and $y \in \mathcal{Y}$.
+**Definition:** Let $X$ and $Y$ be [random variables](/D/rvar) with sets of possible outcomes $\mathcal{X}$ and $\mathcal{Y}$. Then, a joint distribution of $X$ and $Y$ is a [probability distribution](/D/dist) that specifies the probability of the event that $X = x$ and $Y = y$ for each possible combination of $x \in \mathcal{X}$ and $y \in \mathcal{Y}$.
 
 * The joint distribution of two scalar [random variables](/D/rvar) is called a bivariate distribution.
 

D/dist-marg.md

@@ -27,4 +27,4 @@ username: "JoramSoch"
 ---
 
 
-**Definition**: Let $X$ and $Y$ be [random variables](/D/rvar) with sets of possible outcomes $\mathcal{X}$ and $\mathcal{Y}$. Then, the marginal distribution of $X$ is a [probability distribution](/D/dist) that specifies the probability of the event that $X = x$ irrespective of the value of $Y$ for each possible value $x \in \mathcal{X}$. The marginal distribution can be obtained from the [joint distribution](/D/dist-joint) of $X$ and $Y$ using the [law of marginal probability](/D/prob-marg).
+**Definition:** Let $X$ and $Y$ be [random variables](/D/rvar) with sets of possible outcomes $\mathcal{X}$ and $\mathcal{Y}$. Then, the marginal distribution of $X$ is a [probability distribution](/D/dist) that specifies the probability of the event that $X = x$ irrespective of the value of $Y$ for each possible value $x \in \mathcal{X}$. The marginal distribution can be obtained from the [joint distribution](/D/dist-joint) of $X$ and $Y$ using the [law of marginal probability](/D/prob-marg).

D/dist.md

@@ -27,4 +27,4 @@ username: "JoramSoch"
 ---
 
 
-**Definition**: Let $X$ be a [random variable](/D/rvar) with the set of possible outcomes $\mathcal{X}$. Then, a probability distribution of $X$ is a mathematical function that gives the [probabilities](/D/prob) of occurrence of all possible outcomes $x \in \mathcal{X}$ of this random variable.
+**Definition:** Let $X$ be a [random variable](/D/rvar) with the set of possible outcomes $\mathcal{X}$. Then, a probability distribution of $X$ is a mathematical function that gives the [probabilities](/D/prob) of occurrence of all possible outcomes $x \in \mathcal{X}$ of this random variable.

D/exp.md

@@ -27,7 +27,7 @@ username: "JoramSoch"
 ---
 
 
-**Definition**: Let $X$ be a [random variable](/D/rvar). Then, $X$ is said to be exponentially distributed with rate (or, inverse scale) $\lambda$
+**Definition:** Let $X$ be a [random variable](/D/rvar). Then, $X$ is said to be exponentially distributed with rate (or, inverse scale) $\lambda$
 
 $$ \label{eq:exp}
 X \sim \mathrm{Exp}(\lambda) \; ,

D/gam.md

@@ -28,7 +28,7 @@ username: "JoramSoch"
 ---
 
 
-**Definition**: Let $X$ be a [random variable](/D/rvar). Then, $X$ is said to follow a gamma distribution with shape $a$ and rate $b$
+**Definition:** Let $X$ be a [random variable](/D/rvar). Then, $X$ is said to follow a gamma distribution with shape $a$ and rate $b$
 
 $$ \label{eq:gam}
 X \sim \mathrm{Gam}(a, b) \; ,

D/matn.md

@@ -27,7 +27,7 @@ username: "JoramSoch"
 ---
 
 
-**Definition**: Let $X$ be an $n \times p$ [random matrix](/D/rmat). Then, $X$ is said to be matrix-normally distributed with mean $M$, [covariance](/D/covmat) across rows $U$ and [covariance](/D/covmat) across columns $V$
+**Definition:** Let $X$ be an $n \times p$ [random matrix](/D/rmat). Then, $X$ is said to be matrix-normally distributed with mean $M$, [covariance](/D/covmat) across rows $U$ and [covariance](/D/covmat) across columns $V$
 
 $$ \label{eq:matn}
 X \sim \mathcal{MN}(M, U, V) \; ,

D/mom.md

@@ -27,8 +27,16 @@ username: "JoramSoch"
 ---
 
 
-**Definition:** Let $X$ be a [random variable](/D/rvar) and let $n$ be a positive integer. Then, the $n$-th moment of $X$, also called ($n$-th) "raw moment" or "crude moment", is defined as the [expected value](/D/mean) of the $n$-th power of $X$:
+**Definition:** Let $X$ be a [random variable](/D/rvar), let $c$ be a [constant](/D/const) and let $n$ be a positive integer. Then, the $n$-th moment of $X$ about $c$ is defined as the [expected value](/D/mean) of the $n$-th power of $X$ minus $c$:
 
 $$ \label{eq:mom}
-\mu_n' = \mathrm{E}[X^n] \; .
-$$
+\mu_n(c) = \mathrm{E}[(X-c)^n] \; .
+$$
+
+The "$n$-th moment of $X$" may also refer to:
+
+* the $n$-th [raw moment](/D/mom-raw) $\mu_n' = \mu_n(0)$;
+
+* the $n$-th [central moment](/D/mom-cent) $\mu_n = \mu_n(\mu)$;
+
+* the $n$-th [standardized moment](/D/mom-stand) $\mu_n^{*} = \mu_n/\sigma^n$.