Norm, while sitting at the bar, rolls a marble towards the wall and sees how close the marble can get to the wall before it stops rolling. The tosses for a Poisson process having rate $\lambda >0$. The distance from the wall at which a marble, rolled at time t, stops rolling is a random variable with density function $f_t(x)=\frac{2(1+tx)}{2+t}, 0\leq x\leq1$, where $x$ is measured in inches. (a) Assuming these random variables are independent and independent of the Poisson process, find the distribution of $X(t)$, the number of tosses that are within a distance of $\frac{1}{2}$ inch from the wall. (b). Suppose that after a marble is tossed, it is picked up after a random amount of time has passed with cdf G, independently of all other marbles. Find the distribution of $Q(t)$, the total number of marbles on the floor at time t that are within $\frac{1}{2}$ inch from the wall.
I am not sure about my thinking process. I would really appreciate any feedback! Thank you!
For point (a) I am thinking that $X(t)$ is a Poisson process with rate $\lambda \cdot \int_0^t f_s(1/2)ds= \lambda \cdot t$.
For point (b): From (a) we have that the tosses that make the marble land 1/2 inch from the wall form a Poisson process with expectation $\lambda t$. We partition this process into $Q(t)$=the number of such marbles that are still on the floor by time t and $Y(t)$ = the number of such marbles that "left " the floor by time t. Then by the Infinite Server Queue we get that the distribution of $Q(t)$ is Poisson with expectation $E[Q(t)]=\lambda t \int_0^1 1-G(t-s)ds$.