Suppose that Xt is a Poisson process with X1 = 3 and X2 = 10; let Tn = inf{t 0 : Xt = n}. What distribution is best suited to model the random variable T10 - T3?

Question

Suppose that Xt is a Poisson process with X1 = 3 and X2 = 10; let Tn = inf{t > 0 : Xt = n}. What distribution is best suited to model the random variable T10 - T3?

BlackTom AI · Accepted Answer

To understand T10 − T3, think about the Poisson process as a sequence of interarrival times between successive events.
Option by option analysis:
Exponential: An exponential time represents the waiting time for a single event (the first arrival). However, T10 − T3 is the waiting time for 7 more events (from the 3rd to the 10th). Thus it is not just a single exponential, but the sum of seven exponentials, so this option is incomplete.
Normal: The normal distribution is not the natural model for waiting times in a Poisson process, especially for sums of independent exponential interarrival times. The gamma/Erlang form arises from the sum of exponentials, not from a symmetric bell curve.
Brownian Motion: Brownian motion is a continuous-time stochastic process, not a distribution for a finite positive waiting time. It describes paths, not a particular waiting-time random variable, so this is not appropriate for T10 − T3.
Multinomial: Multinomial distributions govern counts across categories in a fixed number of trials, not positive continuous waiting times between events in a Poisson process. It does not model waiting times.
Poisson: The Poisson distribution models counts in a fixed interval, not the waiting time between events. While related, it does not describe the distribution of a positive waiting time directly, so this is not the best model for T10 − T3.
Gamma: In a Poisson process with rate λ, the interarrival times are i.i.d. Exponential(λ). The time to observe 7 more events (from T3 to T10) is the sum of 7 independent Exponential(λ) variables, which follows an Erlang/Gamma distribution with shape parameter 7 and rate λ (or scale 1/λ). This is precisely the distribution that models T10 − T3.

Suppose that Xt is a Poisson process with X1 = 3 and X2 = 10; let Tn = inf{t > 0 : Xt = n}. What distribution is best suited to model the random variable T10 - T3?

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