joint pdf of $(X,X)$ Let $X$ be a continuous random variable with uniform distribution on $[0,1]$, i.e, the probability density function of $X$ is $f(x)=1$ on $[0,1]$. Let $Y=X$. Then what is the joint probability density function $h(x,y)$ of $X$ and $Y$? It seems that $h$ is supported on the line $y=x$, which has $0$ measure. I guess the joint pdf does not exist or equals to Dirac delta function. Can anyone confirm? 

$(X,X)$ does not have a density w.r.t. Lebesgue measure on $\mathbb R^2$. Indeed if $D=\{(x,x), x\in [0,1]\}$, then $P_{(X,X)}(D)=1$ whereas $\lambda_2(D)=0$. 
Besides, $P_{(X,X)}(\{(a,b)\})=0$ for any $(a,b)\in \mathbb R^2$, so the distribution of $(X,X)$ is not discrete either.
