There is a process P that generates instantaneous events X, Y, and Z randomly and independently of each other. Two (or more) events cannot be generated at the same time.
At the end of a custom run of duration T seconds, we know that P generated x X, y Y, and z Z events on the overall.
(1) What is the probabilistic number of occurrences of Y within t seconds of a X?
(2) What is the probabilistic number of occurrences of Y such that it was the immediately followed X and within t seconds?
Even if you refrain from solving, pointers will be welcome. Haven't been into probability for a long while and this is my reframing of a time series problem concerning how the evolution of a particular signal affects the probabilities of the next signal by calculating how things would have been if it was all chance. My answer to (1) will be xyt/T. As to (2), my initial thoughts was to view it as a Poisson process but I think that I am complicating things?
You are asking for Pointers & thoughts , hence I am giving this Post.
There are 2 major Issues (other Issues are minor & I will not go into that)
[[A]] You have to know the Probability Distributions to make calculations.
You have to know the rate(s) of event generation.
[[B]] You want to take "instantaneous events" , but that will make it Continuous Case & require Continuous Probability Distributions.
In a second , we might have infinite number of events , then $x$ , $y$ & $z$ might be infinite.
Alternative Way :
Instead you should make it Discrete & consider the smallest time $\delta t$ within which at most 1 event can occur.
Then we have finite number of event slots in $t$ seconds.
We can scale this up & simplify such that 1 event occurs in 1 second , where the events are : $X$ or $Y$ or $Z$ or $W$ , where $W$ is "no event".
When we want to count events occurring in $T$ seconds , it is $T$ events which is $x+y+z+w$. Now , we want to exclude $W$ events & we also want to count the other events , which might use Combinatorics Concepts.
Your requirement 1 will then be to count $X[Z|W]+Y$ (eg $XZY$ , $XWY$ , $XZWY$ , $XWZY$ )
Your requirement 2 will be to count $X[W]+Y$ (eg $XY$ , $XWY$ , $XWWY$ , $XWWWY$ )
I am using $[Z|W]$ to mean "either $Z$ occurs or $W$ occurs" & $+$ to mean "0 or more occurrences"
We can assume Probability of each event is $P(X)=P(Y)=P(Z)=P(W)=1/4$
If we know more about the events , we can assign other Probability values Eg $P(X)=P(Y)=P(Z)=3/10$ & $P(W)=1/10$
We can then make the necessary calculations to get the necessary numbers.
It will be involving Combinatorics Concepts.