Merge pull request #95 from StatProofBook/master

update to master

Author: JoramSoch

I/DbA.md

@@ -8,7 +8,7 @@ title: "Definition by Author"
 
 - [Sample covariance](/D/cov-samp)
 
-### JoramSoch (169 definitions)
+### JoramSoch (170 definitions)
 
 - [Akaike information criterion](/D/aic)
 - [Alternative hypothesis](/D/h1)
@@ -38,6 +38,7 @@ title: "Definition by Author"
 - [Covariance](/D/cov)
 - [Covariance matrix](/D/covmat)
 - [Critical value](/D/cval)
+- [Cross-covariance matrix](/D/covmat-cross)
 - [Cross-entropy](/D/ent-cross)
 - [Cross-validated log model evidence](/D/cvlme)
 - [Cumulant-generating function](/D/cgf)

I/DbN.md

@@ -182,3 +182,4 @@ title: "Definition by Number"
 | D173 | mse | [Mean squared error](/D/mse) | JoramSoch | 2022-03-27 |
 | D174 | ci | [Confidence interval](/D/ci) | JoramSoch | 2022-03-27 |
 | D175 | nw | [Normal-Wishart distribution](/D/nw) | JoramSoch | 2022-05-14 |
+| D176 | covmat-cross | [Cross-covariance matrix](/D/covmat-cross) | JoramSoch | 2022-09-26 |

I/DbT.md

@@ -42,6 +42,7 @@ title: "Definition by Topic"
 - [Covariance](/D/cov)
 - [Covariance matrix](/D/covmat)
 - [Critical value](/D/cval)
+- [Cross-covariance matrix](/D/covmat-cross)
 - [Cross-entropy](/D/ent-cross)
 - [Cross-validated log model evidence](/D/cvlme)
 - [Cumulant-generating function](/D/cgf)

I/PbA.md

@@ -8,7 +8,7 @@ title: "Proof by Author"
 
 - [Covariance matrix of the multinomial distribution](/P/mult-cov)
 
-### JoramSoch (331 proofs)
+### JoramSoch (336 proofs)
 
 - [Accuracy and complexity for the univariate Gaussian](/P/ug-anc)
 - [Accuracy and complexity for the univariate Gaussian with known variance](/P/ugkv-anc)
@@ -49,6 +49,7 @@ title: "Proof by Author"
 - [Covariance matrix of the categorical distribution](/P/cat-cov)
 - [Covariance matrix of the multivariate normal distribution](/P/mvn-cov)
 - [Covariance of independent random variables](/P/cov-ind)
+- [Covariance of the sum of two random vectors](/P/covmat-sum)
 - [Cross-validated log Bayes factor for the univariate Gaussian with known variance](/P/ugkv-cvlbf)
 - [Cross-validated log model evidence for the univariate Gaussian with known variance](/P/ugkv-cvlme)
 - [Cumulative distribution function in terms of probability density function of a continuous random variable](/P/cdf-pdf)
@@ -153,8 +154,8 @@ title: "Proof by Author"
 - [Maximum likelihood estimation for Dirichlet-distributed data](/P/dir-mle)
 - [Maximum likelihood estimation for multiple linear regression](/P/mlr-mle)
 - [Maximum likelihood estimation for Poisson-distributed data](/P/poiss-mle)
-- [Maximum likelihood estimation for simple linear regression](/P/slr-mle)
 - [Maximum likelihood estimation for simple linear regression](/P/slr-mle2)
+- [Maximum likelihood estimation for simple linear regression](/P/slr-mle)
 - [Maximum likelihood estimation for the general linear model](/P/glm-mle)
 - [Maximum likelihood estimation for the Poisson distribution with exposure values](/P/poissexp-mle)
 - [Maximum likelihood estimation for the univariate Gaussian](/P/ug-mle)
@@ -194,8 +195,8 @@ title: "Proof by Author"
 - [Non-invariance of the differential entropy under change of variables](/P/dent-noninv)
 - [(Non-)Multiplicativity of the expected value](/P/mean-mult)
 - [Non-negativity of the expected value](/P/mean-nonneg)
-- [Non-negativity of the Kullback-Leibler divergence](/P/kl-nonneg)
 - [Non-negativity of the Kullback-Leibler divergence](/P/kl-nonneg2)
+- [Non-negativity of the Kullback-Leibler divergence](/P/kl-nonneg)
 - [Non-negativity of the Shannon entropy](/P/ent-nonneg)
 - [Non-negativity of the variance](/P/var-nonneg)
 - [Non-symmetry of the Kullback-Leibler divergence](/P/kl-nonsymm)
@@ -206,8 +207,8 @@ title: "Proof by Author"
 - [One-sample z-test for independent observations](/P/ugkv-ztest1)
 - [Ordinary least squares for multiple linear regression](/P/mlr-ols)
 - [Ordinary least squares for multiple linear regression](/P/mlr-ols2)
-- [Ordinary least squares for simple linear regression](/P/slr-ols)
 - [Ordinary least squares for simple linear regression](/P/slr-ols2)
+- [Ordinary least squares for simple linear regression](/P/slr-ols)
 - [Ordinary least squares for the general linear model](/P/glm-ols)
 - [Paired t-test for dependent observations](/P/ug-ttestp)
 - [Paired z-test for dependent observations](/P/ugkv-ztestp)
@@ -219,6 +220,7 @@ title: "Proof by Author"
 - [Partition of the log model evidence into accuracy and complexity](/P/lme-anc)
 - [Partition of the mean squared error into bias and variance](/P/mse-bnv)
 - [Partition of variance into expected values](/P/var-mean)
+- [Positive semi-definiteness of the covariance matrix](/P/covmat-psd)
 - [Posterior credibility region against the omnibus null hypothesis for Bayesian linear regression](/P/blr-pcr)
 - [Posterior density is proportional to joint likelihood](/P/post-jl)
 - [Posterior distribution for Bayesian linear regression](/P/blr-post)
@@ -314,8 +316,11 @@ title: "Proof by Author"
 - [Scaling of the covariance matrix upon multiplication with constant matrix](/P/covmat-scal)
 - [Scaling of the variance upon multiplication with a constant](/P/var-scal)
 - [Second central moment is variance](/P/momcent-2nd)
+- [Self-covariance equals variance](/P/cov-var)
 - [Simple linear regression is a special case of multiple linear regression](/P/slr-mlr)
 - [Sums of squares for simple linear regression](/P/slr-sss)
+- [Symmetry of the covariance](/P/cov-symm)
+- [Symmetry of the covariance matrix](/P/covmat-symm)
 - [t-distribution is a special case of multivariate t-distribution](/P/t-mvt)
 - [The p-value follows a uniform distribution under the null hypothesis](/P/pval-h0)
 - [The regression line goes through the center of mass point](/P/slr-comp)
@@ -336,10 +341,10 @@ title: "Proof by Author"
 - [Variance of the normal distribution](/P/norm-var)
 - [Variance of the Poisson distribution](/P/poiss-var)
 - [Variance of the sum of two random variables](/P/var-sum)
-- [Weighted least squares for multiple linear regression](/P/mlr-wls)
 - [Weighted least squares for multiple linear regression](/P/mlr-wls2)
-- [Weighted least squares for simple linear regression](/P/slr-wls)
+- [Weighted least squares for multiple linear regression](/P/mlr-wls)
 - [Weighted least squares for simple linear regression](/P/slr-wls2)
+- [Weighted least squares for simple linear regression](/P/slr-wls)
 - [Weighted least squares for the general linear model](/P/glm-wls)
 
 ### kantundpeterpan (1 proof)
@@ -351,13 +356,15 @@ title: "Proof by Author"
 - [Chi-squared distribution is a special case of gamma distribution](/P/chi2-gam)
 - [Moments of the chi-squared distribution](/P/chi2-mom)
 
-### majapavlo (5 proofs)
+### majapavlo (7 proofs)
 
 - [Cumulative distribution function of the log-normal distribution](/P/lognorm-cdf)
+- [Mean of the log-normal distribution](/P/lognorm-mean)
 - [Median of the log-normal distribution](/P/lognorm-med)
 - [Mode of the log-normal distribution](/P/lognorm-mode)
 - [Probability density function of the log-normal distribution](/P/lognorm-pdf)
 - [Quantile function of the log-normal distribution](/P/lognorm-qf)
+- [Variance of the log-normal distribution](/P/lognorm-var)
 
 ### StatProofBook (1 proof)
 

I/PbN.md

@@ -354,3 +354,10 @@ title: "Proof by Number"
 | P346 | ng-samp | [Sampling from the normal-gamma distribution](/P/ng-samp) | JoramSoch | 2022-09-22 |
 | P347 | covmat-inv | [Invariance of the covariance matrix under addition of constant vector](/P/covmat-inv) | JoramSoch | 2022-09-22 |
 | P348 | covmat-scal | [Scaling of the covariance matrix upon multiplication with constant matrix](/P/covmat-scal) | JoramSoch | 2022-09-22 |
+| P349 | covmat-sum | [Covariance of the sum of two random vectors](/P/covmat-sum) | JoramSoch | 2022-09-26 |
+| P350 | covmat-symm | [Symmetry of the covariance matrix](/P/covmat-symm) | JoramSoch | 2022-09-26 |
+| P351 | covmat-psd | [Positive semi-definiteness of the covariance matrix](/P/covmat-psd) | JoramSoch | 2022-09-26 |
+| P352 | cov-var | [Self-covariance equals variance](/P/cov-var) | JoramSoch | 2022-09-26 |
+| P353 | cov-symm | [Symmetry of the covariance](/P/cov-symm) | JoramSoch | 2022-09-26 |
+| P354 | lognorm-mean | [Mean of the log-normal distribution](/P/lognorm-mean) | majapavlo | 2022-10-02 |
+| P355 | lognorm-var | [Variance of the log-normal distribution](/P/lognorm-var) | majapavlo | 2022-10-02 |

I/PbT.md

@@ -53,6 +53,7 @@ title: "Proof by Topic"
 - [Covariance matrix of the multinomial distribution](/P/mult-cov)
 - [Covariance matrix of the multivariate normal distribution](/P/mvn-cov)
 - [Covariance of independent random variables](/P/cov-ind)
+- [Covariance of the sum of two random vectors](/P/covmat-sum)
 - [Cross-validated log Bayes factor for the univariate Gaussian with known variance](/P/ugkv-cvlbf)
 - [Cross-validated log model evidence for the univariate Gaussian with known variance](/P/ugkv-cvlme)
 - [Cumulative distribution function in terms of probability density function of a continuous random variable](/P/cdf-pdf)
@@ -186,8 +187,8 @@ title: "Proof by Topic"
 - [Maximum likelihood estimation for Dirichlet-distributed data](/P/dir-mle)
 - [Maximum likelihood estimation for multiple linear regression](/P/mlr-mle)
 - [Maximum likelihood estimation for Poisson-distributed data](/P/poiss-mle)
-- [Maximum likelihood estimation for simple linear regression](/P/slr-mle)
 - [Maximum likelihood estimation for simple linear regression](/P/slr-mle2)
+- [Maximum likelihood estimation for simple linear regression](/P/slr-mle)
 - [Maximum likelihood estimation for the general linear model](/P/glm-mle)
 - [Maximum likelihood estimation for the Poisson distribution with exposure values](/P/poissexp-mle)
 - [Maximum likelihood estimation for the univariate Gaussian](/P/ug-mle)
@@ -201,6 +202,7 @@ title: "Proof by Topic"
 - [Mean of the continuous uniform distribution](/P/cuni-mean)
 - [Mean of the exponential distribution](/P/exp-mean)
 - [Mean of the gamma distribution](/P/gam-mean)
+- [Mean of the log-normal distribution](/P/lognorm-mean)
 - [Mean of the matrix-normal distribution](/P/matn-mean)
 - [Mean of the multinomial distribution](/P/mult-mean)
 - [Mean of the multivariate normal distribution](/P/mvn-mean)
@@ -236,8 +238,8 @@ title: "Proof by Topic"
 - [Non-invariance of the differential entropy under change of variables](/P/dent-noninv)
 - [(Non-)Multiplicativity of the expected value](/P/mean-mult)
 - [Non-negativity of the expected value](/P/mean-nonneg)
-- [Non-negativity of the Kullback-Leibler divergence](/P/kl-nonneg)
 - [Non-negativity of the Kullback-Leibler divergence](/P/kl-nonneg2)
+- [Non-negativity of the Kullback-Leibler divergence](/P/kl-nonneg)
 - [Non-negativity of the Shannon entropy](/P/ent-nonneg)
 - [Non-negativity of the variance](/P/var-nonneg)
 - [Non-symmetry of the Kullback-Leibler divergence](/P/kl-nonsymm)
@@ -251,8 +253,8 @@ title: "Proof by Topic"
 - [One-sample z-test for independent observations](/P/ugkv-ztest1)
 - [Ordinary least squares for multiple linear regression](/P/mlr-ols)
 - [Ordinary least squares for multiple linear regression](/P/mlr-ols2)
-- [Ordinary least squares for simple linear regression](/P/slr-ols)
 - [Ordinary least squares for simple linear regression](/P/slr-ols2)
+- [Ordinary least squares for simple linear regression](/P/slr-ols)
 - [Ordinary least squares for the general linear model](/P/glm-ols)
 
 ### P
@@ -267,6 +269,7 @@ title: "Proof by Topic"
 - [Partition of the log model evidence into accuracy and complexity](/P/lme-anc)
 - [Partition of the mean squared error into bias and variance](/P/mse-bnv)
 - [Partition of variance into expected values](/P/var-mean)
+- [Positive semi-definiteness of the covariance matrix](/P/covmat-psd)
 - [Posterior credibility region against the omnibus null hypothesis for Bayesian linear regression](/P/blr-pcr)
 - [Posterior density is proportional to joint likelihood](/P/post-jl)
 - [Posterior distribution for Bayesian linear regression](/P/blr-post)
@@ -376,8 +379,11 @@ title: "Proof by Topic"
 - [Scaling of the covariance matrix upon multiplication with constant matrix](/P/covmat-scal)
 - [Scaling of the variance upon multiplication with a constant](/P/var-scal)
 - [Second central moment is variance](/P/momcent-2nd)
+- [Self-covariance equals variance](/P/cov-var)
 - [Simple linear regression is a special case of multiple linear regression](/P/slr-mlr)
 - [Sums of squares for simple linear regression](/P/slr-sss)
+- [Symmetry of the covariance](/P/cov-symm)
+- [Symmetry of the covariance matrix](/P/covmat-symm)
 
 ### T
 
@@ -402,15 +408,16 @@ title: "Proof by Topic"
 - [Variance of the binomial distribution](/P/bin-var)
 - [Variance of the gamma distribution](/P/gam-var)
 - [Variance of the linear combination of two random variables](/P/var-lincomb)
+- [Variance of the log-normal distribution](/P/lognorm-var)
 - [Variance of the normal distribution](/P/norm-var)
 - [Variance of the Poisson distribution](/P/poiss-var)
 - [Variance of the sum of two random variables](/P/var-sum)
 - [Variance of the Wald distribution](/P/wald-var)
 
 ### W
 
-- [Weighted least squares for multiple linear regression](/P/mlr-wls)
 - [Weighted least squares for multiple linear regression](/P/mlr-wls2)
-- [Weighted least squares for simple linear regression](/P/slr-wls)
+- [Weighted least squares for multiple linear regression](/P/mlr-wls)
 - [Weighted least squares for simple linear regression](/P/slr-wls2)
+- [Weighted least squares for simple linear regression](/P/slr-wls)
 - [Weighted least squares for the general linear model](/P/glm-wls)

I/PwS.md

@@ -55,8 +55,8 @@ title: "Proofs without Source"
 - [Maximum likelihood estimation can result in biased estimates](/P/mle-bias)
 - [Maximum likelihood estimation for multiple linear regression](/P/mlr-mle)
 - [Maximum likelihood estimation for Poisson-distributed data](/P/poiss-mle)
-- [Maximum likelihood estimation for simple linear regression](/P/slr-mle)
 - [Maximum likelihood estimation for simple linear regression](/P/slr-mle2)
+- [Maximum likelihood estimation for simple linear regression](/P/slr-mle)
 - [Maximum likelihood estimation for the general linear model](/P/glm-mle)
 - [Maximum likelihood estimation for the Poisson distribution with exposure values](/P/poissexp-mle)
 - [Mean of the categorical distribution](/P/cat-mean)
@@ -117,6 +117,6 @@ title: "Proofs without Source"
 - [Transposition of a matrix-normal random variable](/P/matn-trans)
 - [Variance of the Wald distribution](/P/wald-var)
 - [Weighted least squares for multiple linear regression](/P/mlr-wls2)
-- [Weighted least squares for simple linear regression](/P/slr-wls)
 - [Weighted least squares for simple linear regression](/P/slr-wls2)
+- [Weighted least squares for simple linear regression](/P/slr-wls)
 - [Weighted least squares for the general linear model](/P/glm-wls)

I/ToC.md

@@ -422,9 +422,11 @@ 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; 3.6.4. **[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>
    &emsp;&ensp; 3.7.1. *[Definition](/D/chi2)* <br>

P/lognorm-mean.md

@@ -0,0 +1,100 @@
+---
+layout: proof
+mathjax: true
+
+author: "Maja Pavlovic"
+affiliation: "Queen Mary University London"
+e_mail: "m.pavlovic@se22.qmul.ac.uk"
+date: 2022-10-02 09:46:00
+
+title: "Mean of the log-normal distribution"
+chapter: "Probability Distributions"
+section: "Univariate continuous distributions"
+topic: "Log-normal distribution"
+theorem: "Mean"
+
+sources:
+  - authors: "Taboga, Marco"
+    year: 2022
+    title: "Log-normal distribution"
+    in: "Lectures on probability theory and mathematical statistics"
+    pages: "retrieved on 2022-10-01"
+    url: "https://www.statlect.com/probability-distributions/log-normal-distribution"
+
+proof_id: "P354"
+shortcut: "lognorm-mean"
+username: "majapavlo"
+---
+
+**Theorem:** Let $X$ be a [random variable](/D/rvar) following a [log-normal distribution](/D/lognorm):
+
+$$ \label{eq:lognorm}
+X \sim \ln  \mathcal{N}(\mu, \sigma^2) 
+$$
+
+Then, the [mean or expected value](/D/mean) of $X$ is
+
+$$
+\mathrm{E}(X) = \exp \left( \mu + \frac{1}{2} \sigma^2 \right) 
+$$
+
+
+**Proof:** The [expected value](/D/mean) is the probability-weighted average over all possible values:
+
+$$ \label{eq:mean}
+\mathrm{E}(X) = \int_{\mathcal{X}} x \cdot f_X(x) \, \mathrm{d}x 
+$$
+
+With the [probability density function of the log-normal distribution](/P/lognorm-pdf), this is:
+
+$$ \label{eq:lognorm-mean-s1}
+\begin{split}
+\mathrm{E}(X) &= \int_{0}^{+\infty} x \cdot \frac{1}{x\sqrt{2 \pi \sigma^2} } \cdot \exp \left[ -\frac{1}{2}  \frac{\left(\ln x-\mu\right)^2}{\sigma^2} \right]  \mathrm{d}x \\
+&= \frac{1}{\sqrt{2 \pi \sigma^2} } \int_{0}^{+\infty} \exp \left[ -\frac{1}{2}  \frac{\left(\ln x-\mu\right)^2}{\sigma^2} \right] \mathrm{d}x
+\end{split}
+$$
+
+Substituting $z = \frac{\ln x -\mu}{\sigma}$, i.e. $x = \exp \left( \mu + \sigma z \right )$, we have:
+
+$$ \label{eq:lognorm-mean-s2}
+\begin{split}
+\mathrm{E}(X) &= \frac{1}{\sqrt{2 \pi \sigma^2} } \int_{(-\infty -\mu )/ (\sigma)}^{(\ln x -\mu )/ (\sigma)} \exp \left( -\frac{1}{2}  z^2 \right) \mathrm{d} \left[ \exp \left( \mu +\sigma z \right) \right] \\
+&= \frac{1}{\sqrt{2 \pi \sigma^2} } \int_{-\infty}^{+\infty} \exp \left( -\frac{1}{2}  z^2 \right) \sigma \exp \left( \mu +\sigma z \right) \mathrm{d}z \\
+&= \frac{1}{\sqrt{2 \pi} } \int_{-\infty}^{+\infty} \exp \left( -\frac{1}{2}  z^2 + \sigma z + \mu \right)  \mathrm{d}z \\
+&= \frac{1}{\sqrt{2 \pi} } \int_{-\infty}^{+\infty} \exp \left[  -\frac{1}{2} \left(  z^2  - 2 \sigma z - 2 \mu \right) \right]  \mathrm{d}z
+\end{split}
+$$
+
+Now multiplying $\exp \left( \frac{1}{2} \sigma^2 \right)$ and $\exp \left( -\frac{1}{2} \sigma^2 \right)$, we have:
+
+$$ \label{eq:lognorm-mean-s3}
+\begin{split}
+\mathrm{E}(X) &= \frac{1}{\sqrt{2 \pi} } \int_{-\infty}^{+\infty} \exp \left[  -\frac{1}{2} \left(  z^2  - 2 \sigma z + \sigma^2 - 2 \mu - \sigma^2 \right) \right]  \mathrm{d}z \\
+&= \frac{1}{\sqrt{2 \pi} } \int_{-\infty}^{+\infty} \exp \left[ -\frac{1}{2} \left( z^2 - 2\sigma z + \sigma^2 \right) \right] \exp \left( \mu + \frac{1}{2} \sigma^2  \right) \mathrm{d}z \\
+&= \exp \left( \mu + \frac{1}{2} \sigma^2  \right) \int_{-\infty}^{+\infty} \frac{1}{\sqrt{2 \pi} } \exp \left[ -\frac{1}{2} \left( z - \sigma \right)^2 \right] \mathrm{d}z
+\end{split}
+$$
+
+The [probability density function of a normal distribution](/P/norm-pdf) is given by
+
+$$ \label{eq:norm-pdf}
+f_X(x) = \frac{1}{\sqrt{2 \pi} \sigma} \cdot \exp \left[ -\frac{1}{2} \left( \frac{x-\mu}{\sigma} \right)^2 \right]
+$$
+
+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), we get
+
+$$ \label{eq:def-pdf}
+\int_{-\infty}^{+\infty} \frac{1}{\sqrt{2 \pi}} \cdot \exp \left[ -\frac{1}{2} \left({x-\mu} \right)^2 \right]  \mathrm{d}x  = 1
+$$
+
+and applying \eqref{eq:def-pdf} to \eqref{eq:lognorm-mean-s3}, we have:
+
+$$ \label{eq:lognorm-mean}
+\mathrm{E}(X) = \exp \left( \mu + \frac{1}{2} \sigma^2  \right) .
+$$