Fix a real number $\lambda > 0$ and consider a Poisson process of rate $\lambda$. Let $T = \{T_1, T_2, \ldots\}$ be the (countable and random) set of arrival times of the process. Let $I$ be a random variable with values in $\{1,2,3,\ldots\}$ and denote $X = T_I$ (the arrival time with index $I$) with continuous probability density $f_X$. Derive an expression for:
a) $\mathbb{P}(X=x_0 \ | \ x_0 \in T)$ in terms of $f_X, \lambda$ and $x_0$.
b) $\mathbb{P}(X=x_0 \ | \ x_0 \in T \mbox{ and } x_1 \in T)$ in terms of $f_X, \lambda, x_0$ and $x_1$.
$\textbf{Remark.}$ The variable $X$ is conditionally on $I$ a discrete random variable, but in general it is a continuous random variable (since the set $T$ of continuously distributed times is also random).
Here is a possible approach for a) (perhaps not the only one):
By continuity we have $\mathbb{P}(X=x_0 \ | \ x_0 \in T) = \lim_{\varepsilon \to 0^{+}} \mathbb{P}(X \in [x_0, x_0 + \varepsilon] \ | \ [x_0,x_0+\varepsilon] \cap T \mbox{ is non-empty})$ which, by the definition of conditional probability, can also be expressed as $$ \lim_{\varepsilon \to 0^{+}} \frac{\mathbb{P}(X\in [x_0, x_0+\varepsilon] \mbox{ and } [x_0,x_0+\varepsilon] \cap T \mbox{ is non-empty})}{\mathbb{P}([x_0,x_0+\varepsilon] \cap T \mbox{ is non-empty})}. $$ Since $X = T_I$ is itself an arrival time, the event $X\in [x_0, x_0+\varepsilon]$ implies the event ``$[x_0,x_0+\varepsilon] \cap T \mbox{ is non-empty}$'' and hence the latter expression equals $$ \lim_{\varepsilon \to 0^{+}} \frac{\mathbb{P}(X\in [x_0, x_0+\varepsilon])}{\mathbb{P}([x_0,x_0+\varepsilon] \cap T \mbox{ is non-empty})}. $$ With $F_X$ denoting, as usual, the cumulative distribution function of $X$, the numerator above equals $F_X(x_0 + \varepsilon) - F_X(x_0)$. The denominator equals $1-e^{-\lambda \varepsilon}$ by the well known property that the number of arrival times in any interval of length $\ell$ of a Poisson process has Poisson$(\lambda\ell)$ distribution (together with $\mathbb{P}(\mbox{Poisson}(s) = 0) = e^{-s}$). It remains to see that (e.g. by using L'Hospital's rule) $$ \lim_{\varepsilon \to 0^{+}} \frac{F_X(x_0 + \varepsilon) - F_X(x_0)}{1-e^{-\lambda \varepsilon}} = \lim_{\varepsilon \to 0^{+}} \frac{f_X(x_0 + \varepsilon)}{\lambda e^{-\lambda\varepsilon}} = \frac{f_X(x_0)}{\lambda} $$ and so the desired probability is $\frac{f_X(x_0)}{\lambda}$.
Any idea if the above (or some other approach) can be used for b)? (I tried also to solve a) by suitably using the law of total probability, but got things which did not make sense.) Any help appreciated!