To calculate the probability P(Z(t-c)=m | Z(t)=k) in a Poisson process Z(t) with rate lambda, one must consider the properties of the process. Given that the jump times are uniformly distributed over the interval, the conditional distribution can be derived using the formula for Poisson probabilities. The relevant probabilities can be expressed as P(Z(t-c)=m) and P(Z(t)=k), applying the conditional probability formula. The solution involves calculating the appropriate factorial terms and using the Poisson distribution's characteristics. Understanding these principles allows for accurate computation of the desired probability in the context of the Poisson process.
