integrated new content

A few minor adjustments to the proofs "lognorm-mean" and "lognorm-var" were performed.

Author: JoramSoch

I/ToC.md

@@ -422,10 +422,10 @@ title: "Table of Contents"
    &emsp;&ensp; 3.6.1. *[Definition](/D/lognorm)* <br>
    &emsp;&ensp; 3.6.2. **[Probability density function](/P/lognorm-pdf)** <br>
    &emsp;&ensp; 3.6.3. **[Cumulative distribution function](/P/lognorm-cdf)** <br>
-   &emsp;&ensp; 3.6.4. **[Median](/P/lognorm-med)** <br>
-   &emsp;&ensp; 3.6.5. **[Mode](/P/lognorm-mode)** <br>
-   &emsp;&ensp; 3.6.6. **[Quantile Function](/P/lognorm-qf)** <br>
-   &emsp;&ensp; 3.6.7. **[Mean](/P/lognorm-mean)** <br>
+   &emsp;&ensp; 3.6.4. **[Quantile Function](/P/lognorm-qf)** <br>
+   &emsp;&ensp; 3.6.5. **[Mean](/P/lognorm-mean)** <br>   
+   &emsp;&ensp; 3.6.6. **[Median](/P/lognorm-med)** <br>
+   &emsp;&ensp; 3.6.7. **[Mode](/P/lognorm-mode)** <br>
    &emsp;&ensp; 3.6.8. **[Variance](/P/lognorm-var)** <br>
 
    3.7. Chi-squared distribution <br>

P/lognorm-mean.md

@@ -54,7 +54,7 @@ $$ \label{eq:lognorm-mean-s1}
 \end{split}
 $$
 
-Substituting $z = \frac{\ln x -\mu}{\sigma}$, i.e. $x = \exp \left( \mu + \sigma z \right )$ we have:
+Substituting $z = \frac{\ln x -\mu}{\sigma}$, i.e. $x = \exp \left( \mu + \sigma z \right )$, we have:
 
 $$ \label{eq:lognorm-mean-s2}
 \begin{split}
@@ -75,19 +75,19 @@ $$ \label{eq:lognorm-mean-s3}
 \end{split}
 $$
 
-The [probability density function of a normal distribution](/P/norm-pdf) reads: 
+The [probability density function of a normal distribution](/P/norm-pdf) is given by
 
 $$ \label{eq:norm-pdf}
 f_X(x) = \frac{1}{\sqrt{2 \pi} \sigma} \cdot \exp \left[ -\frac{1}{2} \left( \frac{x-\mu}{\sigma} \right)^2 \right]
 $$
 
-With unit variance this reads:
+and, with unit variance $\sigma^2 = 1$, this reads:
 
 $$
 = \frac{1}{\sqrt{2 \pi}} \cdot \exp \left[ -\frac{1}{2} \left({x-\mu} \right)^2 \right]
 $$
 
-Using the definition of the [probability density function](/D/pdf)
+Using the definition of the [probability density function](/D/pdf), we get
 
 $$ \label{eq:def-pdf}
 \int_{-\infty}^{+\infty} \frac{1}{\sqrt{2 \pi}} \cdot \exp \left[ -\frac{1}{2} \left({x-\mu} \right)^2 \right]  \mathrm{d}x  = 1

P/lognorm-var.md

@@ -46,17 +46,16 @@ $$ \label{eq:lognorm-var}
 $$
 
 
-**Proof:**
-[Variance](/D/var) is defined as: 
+**Proof:** The [variance](/D/var) of a random variable is defined as
 
 $$ \label{eq:var}
 \mathrm{Var}(X) = \mathrm{E}\left[ (X-\mathrm{E}(X))^2 \right] 
 $$
 
-which, [ partitioned into expected values](/P/var-mean) reads:
+which, [partitioned into expected values](/P/var-mean), reads:
 
 $$ \label{eq:var2}
-\mathrm{Var}(X) = \mathrm{E}\left[ X^2 \right] - \mathrm{E}\left[ X \right]^2
+\mathrm{Var}(X) = \mathrm{E}\left[ X^2 \right] - \mathrm{E}\left[ X \right]^2 \; .
 $$
 
 The [expected value of the log-normal distribution](/P/lognorm-mean) is: 
@@ -95,19 +94,19 @@ $$ \label{eq:second-moment-3}
 \end{split}
 $$
 
-The [probability density function of a normal distribution](/P/norm-pdf) is
+The [probability density function of a normal distribution](/P/norm-pdf) is given by
 
 $$ 
 f_X(x) = \frac{1}{\sqrt{2 \pi} \sigma} \cdot \exp \left[ -\frac{1}{2} \left( \frac{x-\mu}{\sigma} \right)^2 \right]
 $$
 
-with $\mu = 2 \sigma$ and unit variance this reads:
+and, with $\mu = 2 \sigma$ and unit variance, this reads:
 
 $$ 
-= \frac{1}{\sqrt{2 \pi}} \cdot \exp \left[ -\frac{1}{2} \left({x - 2 \sigma} \right)^2 \right]
+= \frac{1}{\sqrt{2 \pi}} \cdot \exp \left[ -\frac{1}{2} \left({x - 2 \sigma} \right)^2 \right] \; .
 $$
 
-Using the definition of the [probability density function](/D/pdf) 
+Using the definition of the [probability density function](/D/pdf), we get
 
 $$ \label{eq:def-pdf}
 \int_{-\infty}^{+\infty} \frac{1}{\sqrt{2 \pi}} \cdot \exp \left[ -\frac{1}{2} \left({x - 2 \sigma} \right)^2 \right]  \mathrm{d}x  = 1 
@@ -119,7 +118,7 @@ $$ \label{eq:second-moment-4}
 \mathrm{E}[X]^2 = \exp \left( 2 \sigma^2 +2 \mu   \right) \; .
 $$
 
-Applying \eqref{eq:second-moment-4} and \eqref{eq:lognorm-mean-ref} to \eqref{eq:var2}, we have:
+Finally, plugging \eqref{eq:second-moment-4} and \eqref{eq:lognorm-mean-ref} into \eqref{eq:var2}, we have:
 
 $$ \label{eq:lognorm-var-2}
 \begin{split}