I have a Poisson process with parameter $\lambda=1$ and arrival times $X_1<X_2<X_3<\cdots$
I round down the arrival times to the nearest hundredth and call it a new process such that $R_1<R_2<R_3<\cdots$ (e.g. if $X_i=4.239,$ then $R_i=4.23$).
Let $T$ denote the steps until the first integer appears in $R_i$.
So for example, if $X_1 = 1.021<X_2=2.438<X_3=4.007$, then $R_1 = 1.02<R_2=2.43<R_3=4.00$ and $T = 3$ since the first integer appears at the third step "$R_3$" and $R_T=4.00$.
How do I calculate $P(R_T=5)$ and also $E[X_T]?$
The way I am thinking about solving this is by using geometric distribution with probability of success = 1/100 (because we will have an integer only if the first two digits after the decimal point are zeros).
But I am not sure if this is the right approach to solve it, and I also wonder how to make use of the fact that we have a Poisson process with parameter $\lambda=1$?
Any help would be appreciated. Thanks.