In a Poisson process with rate \( \lambda \), events occur independently and the number of events in a time interval follows a Poisson distribution. Let \( N(t) \) represent the number of events that occur in the interval \( [0, t] \), and let \( M \) be the number of events in a subinterval of length \( t_0 \) starting at some point in time.
If we’re considering a time interval \([0, t]\) and we want to find the conditional probability mass function (PMF) of \( M \) given \( N(t) = n \), where \( n \) is the total number of events observed in the interval, we can utilize the conditional distribution properties of Poisson processes.
Given that \( N(t) = n \), the counts of events in any disjoint subintervals follow a multinomial distribution. If you split the interval \( [0, t] \) into two parts: the interval \([0, t_0]\) where \( M \) counts the number of events, and the interval \([t_0, t]\) containing the complementary part of the interval, the counts \( M \) and \( N(t) - M \) are conditionally distributed as follows:
\[ (M | N(t) = n) \sim \text{Binomial}(n, p) \]
where \( p = \frac{t_0}{t} \) is the proportion of the total length accounted for by the interval \([0, t_0]\).
The conditional PMF of \( M \) given \( N = n \) can thus be expressed as:
\[ P(M = k | N(t) = n) = \binom{n}{k} p^k (1 - p)^{n-k} \]
for \( k = 0, 1, 2, \ldots, n \), and where \( p = \frac{t_0}{t} \).
This result shows how the number of events in the subinterval is distributed, given a fixed number of total events in the entire interval.