Question

suppose X is uniform on (0,1) and Y=X. Find the joint distribution function of X and Y.

Answer 1

If Y=X then the joint distribution function is \[F_{(X,Y)}(x,y)=\mathbb{P}(X\leq x,Y\leq y)=\mathbb{P}(X\leq\min(x,y))=F_X(\min(x,y))\] which in our case \[F_X(\min(x,y))=\left\{\begin{array}{ll}0&\textrm{, if }x,y\leq 0,\\x&\textrm{, if }0<x\leq y\leq 1,\\y&\textrm{, if }0<y\leq x\leq 1,\\1&\textrm{, if }1<x,y.\end{array}\right.\] The plot of the joint distribution function is attached.