Let $y[n]$ be a random process which is defined as follows: \begin{equation} y[n] = x[n] - x[n-1] \end{equation} where $x[n]$ is iid and takes the values of $\{1,0\}$ with probability $p$ and $1-p$, respectively. I want to find the joint pdf of $y[n]$ and $x[n]$. \begin{equation} f_{y[n],x[n]}(y_n,x_n) = f_{y[n]|x[n]}(y_n|x_n)f_{x[n]}(x_n) \end{equation} \begin{equation} f_{x[n]}(x_n) = p \delta(x_n - 1) + (1-p) \delta(x_n) \end{equation}
I need to find $f_{y[n]|x[n]}(y_n|x_n)$ to proceed. How do I find the conditional pdf?
Edit: If I further want to find $f_{y[n_1],y[n_2]}(y_1,y_2)$ then:
Case $1$: $|n_2 - n_1| = 1$
$f_{y[n_1],y[n_2]}(y_1,y_2)$ = $f_{y[n_2]|y[n_1]}(y_2|y_1)$$f_{y[n_1]}(y_1)$
$y[n_2] = x[n_2] - x[n_2 -1] = x[n_2] - x[n_1] = x[n_2] - (y[n_1] - x[n_1 - 1])$
Since $y[n_1]$ is given, there are two random quantities and the conditional pdf is $p^2\delta(y_2 - (2-y_1)) + (1-p)^2\delta(y_2 + y_1) + 2p(1-p)\delta(y_2 - (1-y_1))$. Is this approach correct?