Calculating the joint pdf of linearly dependent random variables $X$ and $Y=X$

Let $$X$$ and $$Y$$ be two random variables and $$p_{(X,Y)}(x,y)$$ be the joint pdf of $$(X,Y)$$. Suppose that $$(X,Y)$$ transformed to $$(X,X)$$. We want to calculate the joint pdf of transformed random variables. From this, we know that as we normally take $$g(X,Y)=U$$ and $$h(X,Y)=V$$. This gives us that, using the $$g(X,Y)=X=h(X,Y)$$, we get $$U=V$$. Also the corresponding Jacobian matrix can be given as $$\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$$ But this is non-invertible. I have problem in computing the joint pdf of the transformed random variables. I also see online and come to know that the joint pdf of $$(X,X)$$ is $$p_{X}(x)\delta(x-y)$$. Here I am not able to calculate the $$\delta$$ from the usual methods. Also I have calculated the first cdf as $$P(X. From this cdf, can I calculate the desired pdf?

• There is no joint pdf, because this is not a continuous distribution.
– whuber
Commented Jun 28 at 19:21
• @whuber, anyhow can we write the pdf of (X,X) tranformed random variables? Commented Jun 29 at 6:00
• That would be the pdf of $X.$
– whuber
Commented Jun 29 at 17:41

As already explained in comments, there is no joint density in the plane, because all the probability mass of $$(X, X)$$ is concentrated on the diagonal $$y=x$$. There is a density on that diagonal, but that is simply the density of $$X$$.
Maybe you would be better off asking the real problem where this occurs? All probabilities in this setting can be calculated from the density of $$X$$, so it is unclear what is your real problem!
• We have to be a little careful with the meaning of "density on the diagonal," because it depends on how the diagonal is parameterized. For instance, the density of $(X,X)$ relative to Lebesgue measure on the diagonal inherited from $\mathbb R^2$ is $\sqrt{1/2}$ times the density of $X.$