Let $(N_s)$ be a homogene Poisson Process with rate $\lambda>0$ and let $t>0$ be fixed. $T_{N_t}$ is then the last arrival time before time $t$ and $T_{N_t+1}$ is the first arrival time after time $t$. Define the random variables $X_t:=t-T_{N_t}$ and $Y_t:=T_{N_t+1}-t$. Show that the joint distribution of $X_t$ and $Y_t$ is given by $$F(x,y) = \mathbb{P}(X_t\leq x,Y_t\leq y)=((1-e^{-\lambda x}\chi_{(0,t)}(x)+\chi_{[t,\infty)}(x)))(1-e^{-\lambda y})$$ for $x,y>0$. Compute the marginal distributions of $X_t$ and $Y_t$.
First of all, I do not think the distribution is correct, as for large $x$ and $y$ it is bigger than $1$. I think the brackets are in the wrong places. What I did was
\begin{eqnarray} F(x,y) &=& \mathbb{P}(X_t\leq x, Y_t\leq y)\\ &=&\mathbb{P}(t-T_{N_t}\leq x, T_{N_t+1}-t\leq y)\\ &=&\mathbb{P}(T_{N_t}\geq t-x, T_{N_t+1}\leq y+t)\\ &=&\mathbb{P}(T_{N_t}\geq t-x\vert T_{N_t+1}\leq y+t)\mathbb{P}(T_{N_t+1}\leq y+t)\\ &=& (\chi_{(0,t)}(x)\mathbb{P}(T_{N_t}\geq t-x\vert T_{N_t+1}\leq y+t) + \chi_{[t,\infty)}(x)\mathbb{P}(T_{N_t}\geq t-x\vert T_{N_t+1}\leq y+t))\mathbb{P}(N_t+1\leq N_{y+t})\\ &=& (\chi_{(0,t)}(x)\mathbb{P}(N_t\geq N_{t-x}\vert T_{N_t+1}\leq y+t) + \chi_{[t,\infty)}(x))\mathbb{P}(1\leq N_{y})\\ &=& (\chi_{(0,t)}(x)(1-e^{-\lambda x})+\chi_{[t,\infty)}(x))(1-e^{-\lambda y}). \end{eqnarray}
However, I am really unsure whether what I did was correct, especially when it comes to the last 3 equality signs. Also, the distribution is not continuous, which makes me feel kinda uneasy. I would be really happy if someone could clarify whether the last few steps were correct, and if so, why. Also, the marginal distributions are just the first factor of the product for $X_t$ and the second one for $Y_t$, right?
Oh, and on another note, can someone tell me how to make that fat 1 for the characteristic function? Somehow, I couldn't find it out.