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Technically, and perhaps more precisely, one should say each set of finite measure is assigned such a Poisson-distributed random variable.
The Poisson process is one of the most well-known Lévy processes.
A 1-dimensional Poisson process on the interval from 0 to ∞ (essentially this means that the clock starts at time 0; that is when we begin counting) may thus be viewed as an integer-valued nondecreasing random function of time N(t) that counts the number of "arrivals" before time t. Just as a Poisson random variable is characterized by its scalar parameter λ, a Poisson process is characterized by its rate function λ(t), which is the expected number of "events" or "arrivals" that occur per unit time. A homogeneous Poisson process has a constant rate function λ(t) = λ. If the rate remains constant, then the number N(t) of arrivals before time t distribution has a Poisson distribution with expected value λt.
Let Xt be the number of arrivals before time t. Let Tx be the time of the xth arrival, for x = 1, 2, 3, ... . (We are using capital X and capital T for random variables, and lower-case x and lower-case t for non-random quantities.) The random variable Xt has a discrete probability distribution -- a Poisson distribution -- and the random variable Tx has a continuous probability distribution.
Clearly the number of arrivals before time t is less than x if and only if the waiting time until the xth arrival is more than t. In symbols, the event [ Xt < x ] occurs if and only if the event [ Tx > t ]. Consequently the probabilities of these events are the same:
This fact plus knowledge of the Poisson distribution enables us to find the probability distribution of these continuous random variables. In case the rate, i.e., the expected number of arrivals per unit time, remains constant, this is fairly simple. In particular, consider the waiting time until the first arrival. Clearly that time is more than t if and only if the number of arrivals before time t is a 0. If the rate is λ arrivals per unit time, then we have
Consequently, the waiting time until the first arrival has a exponential distribution. This exponential distribution has expected value 1/λ. In other words, if the average rate of arrivals is, for example 6 per minute, then the average waiting time until the first arrival is (unsurprisingly) 1/6 minute. This exponential distribution is memoryless, i.e. we have
This says that the conditional probability that we need to wait, for example, more than another 10 seconds before the first arrival, given that the first arrival has not yet happened after 30 seconds, is no different from the initial probability that we need to wait more than 10 seconds for the first arrival. This is often misunderstood by students taking courses on probability: the fact that P(T1 > 40 | T1 > 30) = P(T1 > 10) does not mean that the events T1 > 40 and T1 > 10 are independent. To summarize: "memorylessness" of the probability distribution of the waiting time T1 until the first arrival means
It does not mean
(That would be independence. These two events are not independent.)
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