rectilinear restrictions to joint distribution, exponentiation.

.Rprofile

@@ -130,6 +130,6 @@ if (knitr::is_latex_output()) {
     text_top_count <- ""
     text_end_count <- ""
 } else {
-    text_top_count <- "[![HitCount](http://hits.dwyl.com/fanwangecon/R4Econ.svg)](https://github.com/FanWangEcon/R4Econ)  [![Star](https://img.shields.io/github/stars/fanwangecon/R4Econ?style=social)](https://github.com/FanWangEcon/R4Econ/stargazers) [![Fork](https://img.shields.io/github/forks/fanwangecon/R4Econ?style=social)](https://github.com/FanWangEcon/R4Econ/network/members) [![Star](https://img.shields.io/github/watchers/fanwangecon/R4Econ?style=social)](https://github.com/FanWangEcon/R4Econ/watchers) [![DOI](https://zenodo.org/badge/173583807.svg)](https://zenodo.org/badge/latestdoi/173583807)"
+    text_top_count <- "[![Star](https://img.shields.io/github/stars/fanwangecon/R4Econ?style=social)](https://github.com/FanWangEcon/R4Econ/stargazers) [![Fork](https://img.shields.io/github/forks/fanwangecon/R4Econ?style=social)](https://github.com/FanWangEcon/R4Econ/network/members) [![Star](https://img.shields.io/github/watchers/fanwangecon/R4Econ?style=social)](https://github.com/FanWangEcon/R4Econ/watchers) [![DOI](https://zenodo.org/badge/173583807.svg)](https://zenodo.org/badge/latestdoi/173583807)"
     text_end_count <- "[![](https://img.shields.io/github/last-commit/fanwangecon/R4Econ)](https://github.com/FanWangEcon/R4Econ/commits/master) [![](https://img.shields.io/github/commit-activity/m/fanwangecon/R4Econ)](https://github.com/FanWangEcon/R4Econ/graphs/commit-activity) [![](https://img.shields.io/github/issues/fanwangecon/R4Econ)](https://github.com/FanWangEcon/R4Econ/issues) [![](https://img.shields.io/github/issues-pr/fanwangecon/R4Econ)](https://github.com/FanWangEcon/R4Econ/pulls)"
 }

README.md

@@ -1,4 +1,4 @@
-[![HitCount](http://hits.dwyl.com/fanwangecon/R4Econ.svg)](https://github.com/FanWangEcon/R4Econ)  [![Star](https://img.shields.io/github/stars/fanwangecon/R4Econ?style=social)](https://github.com/FanWangEcon/R4Econ/stargazers) [![Fork](https://img.shields.io/github/forks/fanwangecon/R4Econ?style=social)](https://github.com/FanWangEcon/R4Econ/network/members) [![Star](https://img.shields.io/github/watchers/fanwangecon/R4Econ?style=social)](https://github.com/FanWangEcon/R4Econ/watchers) [![DOI](https://zenodo.org/badge/173583807.svg)](https://zenodo.org/badge/latestdoi/173583807)
+[![Star](https://img.shields.io/github/stars/fanwangecon/R4Econ?style=social)](https://github.com/FanWangEcon/R4Econ/stargazers) [![Fork](https://img.shields.io/github/forks/fanwangecon/R4Econ?style=social)](https://github.com/FanWangEcon/R4Econ/network/members) [![Star](https://img.shields.io/github/watchers/fanwangecon/R4Econ?style=social)](https://github.com/FanWangEcon/R4Econ/watchers) [![DOI](https://zenodo.org/badge/173583807.svg)](https://zenodo.org/badge/latestdoi/173583807)
 
 This is a work-in-progress [website](https://fanwangecon.github.io/R4Econ/) consisting of R panel data and optimization examples for Statistics/Econometrics/Economic Analysis.
 
@@ -252,11 +252,15 @@ Please contact [FanWangEcon](https://fanwangecon.github.io/) for issues or probl
 
 # 8  Mathematics
 
-## 8.1  Analytical Solutions
+## 8.1  Basics
 
 1. [Quadratic and other Rescaling of Parameters with Fixed Min and Max](https://fanwangecon.github.io/R4Econ/math/solutions/htmlpdfr/fs_rescale.html): [**rmd**](https://github.com/FanWangEcon/R4Econ/blob/master/math/solutions//fs_rescale.Rmd) \| [**r**](https://github.com/FanWangEcon/R4Econ/blob/master/math/solutions/htmlpdfr/fs_rescale.R) \| [**pdf**](https://github.com/FanWangEcon/R4Econ/blob/master/math/solutions/htmlpdfr/fs_rescale.pdf) \| [**html**](https://fanwangecon.github.io/R4Econ/math/solutions/htmlpdfr/fs_rescale.html)
 	+ Given a < x < b, use f(x) to rescale x, such that f(a)=a, f(b)=b, but f(z)=0.5*z for some z between a and b. Solve using the quadratic function with three equations and three unknowns uniquely. 
-2. [linear solve x with f(x) = 0](https://fanwangecon.github.io/R4Econ/math/solutions/htmlpdfr/fs_solu_x_lin.html): [**rmd**](https://github.com/FanWangEcon/R4Econ/blob/master/math/solutions//fs_solu_x_lin.Rmd) \| [**r**](https://github.com/FanWangEcon/R4Econ/blob/master/math/solutions/htmlpdfr/fs_solu_x_lin.R) \| [**pdf**](https://github.com/FanWangEcon/R4Econ/blob/master/math/solutions/htmlpdfr/fs_solu_x_lin.pdf) \| [**html**](https://fanwangecon.github.io/R4Econ/math/solutions/htmlpdfr/fs_solu_x_lin.html)
+2. [Positive and Negative Exponents x is Below or Above 1](https://fanwangecon.github.io/R4Econ/math/solutions/htmlpdfr/fs_exponents.html): [**rmd**](https://github.com/FanWangEcon/R4Econ/blob/master/math/solutions//fs_exponents.Rmd) \| [**r**](https://github.com/FanWangEcon/R4Econ/blob/master/math/solutions/htmlpdfr/fs_exponents.R) \| [**pdf**](https://github.com/FanWangEcon/R4Econ/blob/master/math/solutions/htmlpdfr/fs_exponents.pdf) \| [**html**](https://fanwangecon.github.io/R4Econ/math/solutions/htmlpdfr/fs_exponents.html)
+	+ Positive exponentials are strictly increasing. Negative exponentials are strictly decreasing. 
+	+ If 0 < a < b, 0 < a^(|c|) < b^(|c|), and a^(-|c|) > b^(-|c|) > 0.
+	+ A positive number below 1 to a negative exponents is above 1, and a positive number above 1 to a negative exponents is below 1.  
+3. [linear solve x with f(x) = 0](https://fanwangecon.github.io/R4Econ/math/solutions/htmlpdfr/fs_solu_x_lin.html): [**rmd**](https://github.com/FanWangEcon/R4Econ/blob/master/math/solutions//fs_solu_x_lin.Rmd) \| [**r**](https://github.com/FanWangEcon/R4Econ/blob/master/math/solutions/htmlpdfr/fs_solu_x_lin.R) \| [**pdf**](https://github.com/FanWangEcon/R4Econ/blob/master/math/solutions/htmlpdfr/fs_solu_x_lin.pdf) \| [**html**](https://fanwangecon.github.io/R4Econ/math/solutions/htmlpdfr/fs_solu_x_lin.html)
 	+ Evaluate and solve statistically relevant problems with one equation and one unknown that permit analytical solutions.
 
 ## 8.2  Inequality Models
@@ -288,6 +292,15 @@ Please contact [FanWangEcon](https://fanwangecon.github.io/) for issues or probl
 	+ Random Sampling (Monte Carlo) integrate shocks.
 	+ Trapezoidal rule (symmetric rectangles) integrate normal shock.
 
+## 9.2  Discrete Random Variable
+
+1. [Obtaining Joint Distribution from Marginal with Rectilinear Restrictions](https://fanwangecon.github.io/R4Econ/statistics/discrandvar/htmlpdfr/fs_discrandvar_marg2joint.html): [**rmd**](https://github.com/FanWangEcon/R4Econ/blob/master/statistics/discrandvar//fs_discrandvar_marg2joint.Rmd) \| [**r**](https://github.com/FanWangEcon/R4Econ/blob/master/statistics/discrandvar/htmlpdfr/fs_discrandvar_marg2joint.R) \| [**pdf**](https://github.com/FanWangEcon/R4Econ/blob/master/statistics/discrandvar/htmlpdfr/fs_discrandvar_marg2joint.pdf) \| [**html**](https://fanwangecon.github.io/R4Econ/statistics/discrandvar/htmlpdfr/fs_discrandvar_marg2joint.html)
+	+ Solve for joint distributional mass given marginal distributional mass given rectilinear assumptions.
+	+ **r**: *qr()*
+2. [Obtaining Joint Distribution from Conditional with Rectilinear Restrictions](https://fanwangecon.github.io/R4Econ/statistics/discrandvar/htmlpdfr/fs_discrandvar_condi2joint.html): [**rmd**](https://github.com/FanWangEcon/R4Econ/blob/master/statistics/discrandvar//fs_discrandvar_condi2joint.Rmd) \| [**r**](https://github.com/FanWangEcon/R4Econ/blob/master/statistics/discrandvar/htmlpdfr/fs_discrandvar_condi2joint.R) \| [**pdf**](https://github.com/FanWangEcon/R4Econ/blob/master/statistics/discrandvar/htmlpdfr/fs_discrandvar_condi2joint.pdf) \| [**html**](https://fanwangecon.github.io/R4Econ/statistics/discrandvar/htmlpdfr/fs_discrandvar_condi2joint.html)
+	+ Solve for joint distributional mass given conditional distributional mass given rectilinear assumptions.
+	+ **r**: *qr() + solve() + matrix()*
+
 # 10  Tables and Graphs
 
 ## 10.1  R Base Plots
@@ -309,10 +322,11 @@ Please contact [FanWangEcon](https://fanwangecon.github.io/) for issues or probl
 1. [ggplot Scatter Plot Three Continuous Variables and Multiple Categorical Variables](https://fanwangecon.github.io/R4Econ/tabgraph/ggscatter/htmlpdfr/fs_ggscatter_3cts_mdisc.html): [**rmd**](https://github.com/FanWangEcon/R4Econ/blob/master/tabgraph/ggscatter//fs_ggscatter_3cts_mdisc.Rmd) \| [**r**](https://github.com/FanWangEcon/R4Econ/blob/master/tabgraph/ggscatter/htmlpdfr/fs_ggscatter_3cts_mdisc.R) \| [**pdf**](https://github.com/FanWangEcon/R4Econ/blob/master/tabgraph/ggscatter/htmlpdfr/fs_ggscatter_3cts_mdisc.pdf) \| [**html**](https://fanwangecon.github.io/R4Econ/tabgraph/ggscatter/htmlpdfr/fs_ggscatter_3cts_mdisc.html)
 	+ Two continuous variables for the x-axis and the y-axis, another continuous variable for size of scatter, other categorical variables for scatter shape and size.
 	+ **ggplot**: *ggplot() + geom_jitter() + geom_smooth() + scale_colour_manual() + scale_shape_discrete() + scale_linetype_manual() + scale_x_continuous() + scale_y_continuous() + theme_bw() + theme()*
-2. [ggplot Scatter Points Facet Wrap Over Categories to Generate Subplots](https://fanwangecon.github.io/R4Econ/tabgraph/ggscatter/htmlpdfr/fs_ggscatter_facet_wrap.html): [**rmd**](https://github.com/FanWangEcon/R4Econ/blob/master/tabgraph/ggscatter//fs_ggscatter_facet_wrap.Rmd) \| [**r**](https://github.com/FanWangEcon/R4Econ/blob/master/tabgraph/ggscatter/htmlpdfr/fs_ggscatter_facet_wrap.R) \| [**pdf**](https://github.com/FanWangEcon/R4Econ/blob/master/tabgraph/ggscatter/htmlpdfr/fs_ggscatter_facet_wrap.pdf) \| [**html**](https://fanwangecon.github.io/R4Econ/tabgraph/ggscatter/htmlpdfr/fs_ggscatter_facet_wrap.html)
+2. [ggplot Multiple Scatter-Lines and Facet Wrap Over Categories](https://fanwangecon.github.io/R4Econ/tabgraph/ggscatter/htmlpdfr/fs_ggscatter_facet_wrap.html): [**rmd**](https://github.com/FanWangEcon/R4Econ/blob/master/tabgraph/ggscatter//fs_ggscatter_facet_wrap.Rmd) \| [**r**](https://github.com/FanWangEcon/R4Econ/blob/master/tabgraph/ggscatter/htmlpdfr/fs_ggscatter_facet_wrap.R) \| [**pdf**](https://github.com/FanWangEcon/R4Econ/blob/master/tabgraph/ggscatter/htmlpdfr/fs_ggscatter_facet_wrap.pdf) \| [**html**](https://fanwangecon.github.io/R4Econ/tabgraph/ggscatter/htmlpdfr/fs_ggscatter_facet_wrap.html)
+	+ ggplot multiple lines with scatter as points and connecting lines.
 	+ Facet wrap to generate subfigures for sub-categories.
 	+ Generate separate plots from data saved separately.
-	+ **r**: *lapply*
+	+ **r**: *apply*
 	+ **ggplot**: *facet_wrap() + geom_smooth() + geom_point() + facet_wrap() + scale_colour_manual() + scale_shape_manual() + scale_linetype_manual()*
 
 ## 10.4  Write and Read Plots

README_appendix.md

@@ -240,11 +240,15 @@
 
 ## Mathematics links
 
-### [Section 8.1 Analytical Solutions][Analytical Solutions] links
+### [Section 8.1 Basics][Basics] links
 
 1. [Quadratic and other Rescaling of Parameters with Fixed Min and Max](https://fanwangecon.github.io/R4Econ/math/solutions/htmlpdfr/fs_rescale.html): [**rmd**](https://github.com/FanWangEcon/R4Econ/blob/master/math/solutions//fs_rescale.Rmd) \| [**r**](https://github.com/FanWangEcon/R4Econ/blob/master/math/solutions/htmlpdfr/fs_rescale.R) \| [**pdf**](https://github.com/FanWangEcon/R4Econ/blob/master/math/solutions/htmlpdfr/fs_rescale.pdf) \| [**html**](https://fanwangecon.github.io/R4Econ/math/solutions/htmlpdfr/fs_rescale.html)
 	+ Given a < x < b, use f(x) to rescale x, such that f(a)=a, f(b)=b, but f(z)=0.5*z for some z between a and b. Solve using the quadratic function with three equations and three unknowns uniquely. 
-2. [linear solve x with f(x) = 0](https://fanwangecon.github.io/R4Econ/math/solutions/htmlpdfr/fs_solu_x_lin.html): [**rmd**](https://github.com/FanWangEcon/R4Econ/blob/master/math/solutions//fs_solu_x_lin.Rmd) \| [**r**](https://github.com/FanWangEcon/R4Econ/blob/master/math/solutions/htmlpdfr/fs_solu_x_lin.R) \| [**pdf**](https://github.com/FanWangEcon/R4Econ/blob/master/math/solutions/htmlpdfr/fs_solu_x_lin.pdf) \| [**html**](https://fanwangecon.github.io/R4Econ/math/solutions/htmlpdfr/fs_solu_x_lin.html)
+2. [Positive and Negative Exponents x is Below or Above 1](https://fanwangecon.github.io/R4Econ/math/solutions/htmlpdfr/fs_exponents.html): [**rmd**](https://github.com/FanWangEcon/R4Econ/blob/master/math/solutions//fs_exponents.Rmd) \| [**r**](https://github.com/FanWangEcon/R4Econ/blob/master/math/solutions/htmlpdfr/fs_exponents.R) \| [**pdf**](https://github.com/FanWangEcon/R4Econ/blob/master/math/solutions/htmlpdfr/fs_exponents.pdf) \| [**html**](https://fanwangecon.github.io/R4Econ/math/solutions/htmlpdfr/fs_exponents.html)
+	+ Positive exponentials are strictly increasing. Negative exponentials are strictly decreasing. 
+	+ If 0 < a < b, 0 < a^(|c|) < b^(|c|), and a^(-|c|) > b^(-|c|) > 0.
+	+ A positive number below 1 to a negative exponents is above 1, and a positive number above 1 to a negative exponents is below 1.  
+3. [linear solve x with f(x) = 0](https://fanwangecon.github.io/R4Econ/math/solutions/htmlpdfr/fs_solu_x_lin.html): [**rmd**](https://github.com/FanWangEcon/R4Econ/blob/master/math/solutions//fs_solu_x_lin.Rmd) \| [**r**](https://github.com/FanWangEcon/R4Econ/blob/master/math/solutions/htmlpdfr/fs_solu_x_lin.R) \| [**pdf**](https://github.com/FanWangEcon/R4Econ/blob/master/math/solutions/htmlpdfr/fs_solu_x_lin.pdf) \| [**html**](https://fanwangecon.github.io/R4Econ/math/solutions/htmlpdfr/fs_solu_x_lin.html)
 	+ Evaluate and solve statistically relevant problems with one equation and one unknown that permit analytical solutions.
 
 ### [Section 8.2 Inequality Models][Inequality Models] links
@@ -276,6 +280,15 @@
 	+ Random Sampling (Monte Carlo) integrate shocks.
 	+ Trapezoidal rule (symmetric rectangles) integrate normal shock.
 
+### [Section 9.2 Discrete Random Variable][Discrete Random Variable] links
+
+1. [Obtaining Joint Distribution from Marginal with Rectilinear Restrictions](https://fanwangecon.github.io/R4Econ/statistics/discrandvar/htmlpdfr/fs_discrandvar_marg2joint.html): [**rmd**](https://github.com/FanWangEcon/R4Econ/blob/master/statistics/discrandvar//fs_discrandvar_marg2joint.Rmd) \| [**r**](https://github.com/FanWangEcon/R4Econ/blob/master/statistics/discrandvar/htmlpdfr/fs_discrandvar_marg2joint.R) \| [**pdf**](https://github.com/FanWangEcon/R4Econ/blob/master/statistics/discrandvar/htmlpdfr/fs_discrandvar_marg2joint.pdf) \| [**html**](https://fanwangecon.github.io/R4Econ/statistics/discrandvar/htmlpdfr/fs_discrandvar_marg2joint.html)
+	+ Solve for joint distributional mass given marginal distributional mass given rectilinear assumptions.
+	+ **r**: *qr()*
+2. [Obtaining Joint Distribution from Conditional with Rectilinear Restrictions](https://fanwangecon.github.io/R4Econ/statistics/discrandvar/htmlpdfr/fs_discrandvar_condi2joint.html): [**rmd**](https://github.com/FanWangEcon/R4Econ/blob/master/statistics/discrandvar//fs_discrandvar_condi2joint.Rmd) \| [**r**](https://github.com/FanWangEcon/R4Econ/blob/master/statistics/discrandvar/htmlpdfr/fs_discrandvar_condi2joint.R) \| [**pdf**](https://github.com/FanWangEcon/R4Econ/blob/master/statistics/discrandvar/htmlpdfr/fs_discrandvar_condi2joint.pdf) \| [**html**](https://fanwangecon.github.io/R4Econ/statistics/discrandvar/htmlpdfr/fs_discrandvar_condi2joint.html)
+	+ Solve for joint distributional mass given conditional distributional mass given rectilinear assumptions.
+	+ **r**: *qr() + solve() + matrix()*
+
 ## Tables and Graphs links
 
 ### [Section 10.1 R Base Plots][R Base Plots] links
@@ -297,10 +310,11 @@
 1. [ggplot Scatter Plot Three Continuous Variables and Multiple Categorical Variables](https://fanwangecon.github.io/R4Econ/tabgraph/ggscatter/htmlpdfr/fs_ggscatter_3cts_mdisc.html): [**rmd**](https://github.com/FanWangEcon/R4Econ/blob/master/tabgraph/ggscatter//fs_ggscatter_3cts_mdisc.Rmd) \| [**r**](https://github.com/FanWangEcon/R4Econ/blob/master/tabgraph/ggscatter/htmlpdfr/fs_ggscatter_3cts_mdisc.R) \| [**pdf**](https://github.com/FanWangEcon/R4Econ/blob/master/tabgraph/ggscatter/htmlpdfr/fs_ggscatter_3cts_mdisc.pdf) \| [**html**](https://fanwangecon.github.io/R4Econ/tabgraph/ggscatter/htmlpdfr/fs_ggscatter_3cts_mdisc.html)
 	+ Two continuous variables for the x-axis and the y-axis, another continuous variable for size of scatter, other categorical variables for scatter shape and size.
 	+ **ggplot**: *ggplot() + geom_jitter() + geom_smooth() + scale_colour_manual() + scale_shape_discrete() + scale_linetype_manual() + scale_x_continuous() + scale_y_continuous() + theme_bw() + theme()*
-2. [ggplot Scatter Points Facet Wrap Over Categories to Generate Subplots](https://fanwangecon.github.io/R4Econ/tabgraph/ggscatter/htmlpdfr/fs_ggscatter_facet_wrap.html): [**rmd**](https://github.com/FanWangEcon/R4Econ/blob/master/tabgraph/ggscatter//fs_ggscatter_facet_wrap.Rmd) \| [**r**](https://github.com/FanWangEcon/R4Econ/blob/master/tabgraph/ggscatter/htmlpdfr/fs_ggscatter_facet_wrap.R) \| [**pdf**](https://github.com/FanWangEcon/R4Econ/blob/master/tabgraph/ggscatter/htmlpdfr/fs_ggscatter_facet_wrap.pdf) \| [**html**](https://fanwangecon.github.io/R4Econ/tabgraph/ggscatter/htmlpdfr/fs_ggscatter_facet_wrap.html)
+2. [ggplot Multiple Scatter-Lines and Facet Wrap Over Categories](https://fanwangecon.github.io/R4Econ/tabgraph/ggscatter/htmlpdfr/fs_ggscatter_facet_wrap.html): [**rmd**](https://github.com/FanWangEcon/R4Econ/blob/master/tabgraph/ggscatter//fs_ggscatter_facet_wrap.Rmd) \| [**r**](https://github.com/FanWangEcon/R4Econ/blob/master/tabgraph/ggscatter/htmlpdfr/fs_ggscatter_facet_wrap.R) \| [**pdf**](https://github.com/FanWangEcon/R4Econ/blob/master/tabgraph/ggscatter/htmlpdfr/fs_ggscatter_facet_wrap.pdf) \| [**html**](https://fanwangecon.github.io/R4Econ/tabgraph/ggscatter/htmlpdfr/fs_ggscatter_facet_wrap.html)
+	+ ggplot multiple lines with scatter as points and connecting lines.
 	+ Facet wrap to generate subfigures for sub-categories.
 	+ Generate separate plots from data saved separately.
-	+ **r**: *lapply*
+	+ **r**: *apply*
 	+ **ggplot**: *facet_wrap() + geom_smooth() + geom_point() + facet_wrap() + scale_colour_manual() + scale_shape_manual() + scale_linetype_manual()*
 
 ### [Section 10.4 Write and Read Plots][Write and Read Plots] links

README_pre.md

@@ -1,4 +1,4 @@
-[![HitCount](http://hits.dwyl.com/fanwangecon/R4Econ.svg)](https://github.com/FanWangEcon/R4Econ)  [![Star](https://img.shields.io/github/stars/fanwangecon/R4Econ?style=social)](https://github.com/FanWangEcon/R4Econ/stargazers) [![Fork](https://img.shields.io/github/forks/fanwangecon/R4Econ?style=social)](https://github.com/FanWangEcon/R4Econ/network/members) [![Star](https://img.shields.io/github/watchers/fanwangecon/R4Econ?style=social)](https://github.com/FanWangEcon/R4Econ/watchers) [![DOI](https://zenodo.org/badge/173583807.svg)](https://zenodo.org/badge/latestdoi/173583807)
+[![Star](https://img.shields.io/github/stars/fanwangecon/R4Econ?style=social)](https://github.com/FanWangEcon/R4Econ/stargazers) [![Fork](https://img.shields.io/github/forks/fanwangecon/R4Econ?style=social)](https://github.com/FanWangEcon/R4Econ/network/members) [![Star](https://img.shields.io/github/watchers/fanwangecon/R4Econ?style=social)](https://github.com/FanWangEcon/R4Econ/watchers) [![DOI](https://zenodo.org/badge/173583807.svg)](https://zenodo.org/badge/latestdoi/173583807)
 
 This is a work-in-progress [website](https://fanwangecon.github.io/R4Econ/) consisting of R panel data and optimization examples for Statistics/Econometrics/Economic Analysis.