$\huge{Part 1 : Theoretical}$

${M = { \{(x,y) \in \mathbb{R}^2 ; 0<x<y<1} \} }$

So we can calculate the Area of ${M}$

$Area(M) = \int_{0}^{1}\int_{x}^{1}\mathbb{1}dy dx=\int_{0}^{1} \Big|y\Big|_x^1dx=\int_{0}^{1}(1-x)dx=\Big|x-\tfrac{x^2}{2}\Big|_0^1dx=1- \tfrac{1}{2}=\tfrac{1}{2}$

We pose $Z=(X,Y)$ and thus $z=(x,y)$

$f_z(z)=\int_{0}^{1}\tfrac{1}{\tfrac{1}{2}}\mathbb{1}_{\{0\le y \le 1 \}.\{0\le x \le y \}}dz = \int_{0}^{1}2.\mathbb{1}_{\{0\le y \le 1 \}.\{0\le x \le y \}}dz = \left\{\begin{matrix}0 \:\: if \:\: z \notin M \\2 \:\: elif\end{matrix}\right.$

Marginal densities calcule :

$f_x(x)=\int_{x}^{1}f_z(z)dy = \left\{\begin{matrix}0 \; if \: x \notin (0,y) \\2.\int_{x}^{1}\mathbb{1}dy=2(1-x) \:\: elif\end{matrix}\right.$

$f_y(y)=\int_{0}^{y}f_z(z)dx = \left\{\begin{matrix}0 \; if \: y \notin (x,1) \\2.\int_{0}^{y}\mathbb{1}dy=2y \:\: elif\end{matrix}\right.$

Independance or not ?

If $x$ and $y$ are independante : $F_{xy}(x,y)=F_x(x).F_y(y)$

We can calculate the 3 :

$F_x(x)=\int f_x(x)dx=\int2(1-x)dx=x(2-x)$

$F_y(y)=\int f_y(y).dy=\int2y.dy=y^{2}$

$F_{xy}(x,y)=\int\int f_{xy}(x,y).dx.dy=\int\int 2.dx.dy=\int2x.dy=2xy$

We can conclude : $F_{xy}(x,y) \neq F_x(x).F_y(y)$

Tha variables $x$ and $y$ are not independante.

$\huge{Part \: 2 : Practical}$

create a sample :

x<-rep(NA,100000)
y<-rep(NA,100000)

for(i in 1:100000){
  t<-runif(2)
  x[i]<-min(t) #because x<y
  y[i]<-max(t) #because y>x
}
z<-data.frame(cbind(x,y))

$F_x(x) :$

vertical <- seq(0,1,length.out =100000 )
plot(sort(x),vertical,type="l",xlab="x",main="repartion function of x")
lines(vertical,vertical*(2-vertical),type="l",col="red")

The two curves merge, we can conclude the accuracy of the formula. $F_x(x)=x(2-x)$ so $f_x(x)=2-2x$

$F_y(y) :$

plot(sort(y),vertical,type="l",xlab="y",main="repartion function of y")
lines(vertical,vertical^2,type="l",col="red")

The two curves merge, we can conclude the accuracy of the formula. $F_y(y)=y^{2}$ so $f_y(y)=2y$

$f_z(z) :$

library(ggplot2)
ggplot(z, aes(x=x, y=y) ) +
  stat_density_2d(aes(fill = ..level..), geom = "polygon")

We can see that worth zero if $z \notin M$ and $=2 \:\: if \:z \in M$