Poisson過程

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-{T|zh-cn:泊松过程;zh-tw:卜瓦松過程;zh-hk:泊松過程}- Poisson-{zh-cn:过程;zh-tw:過程;zh-hk:過程}-（Poisson process，-{zh-cn:大陆译泊松过程、普阿松过程等，台译卜瓦松過程、布瓦松過程、布阿松過程、波以松過程、卜氏過程等;zh-tw:也譯為布瓦松過程、布阿松過程、波以松過程、卜氏過程等}-），是以法國數學家 [[-{zh-cn:泊松;zh-tw:卜瓦松;zh-hk:泊松}-]]（1781 - 1840）的名字命名的。-{zh-cn:泊松;zh-tw:卜瓦松;zh-hk:泊松}-過程是隨機過程的一種，是以事件的發生時間來定義的。我們說一個隨機過程 N(t) 是一個時間齊次的一維-{zh-cn:泊松;zh-tw:卜瓦松;zh-hk:泊松}-過程，如果它滿足以下條件：

在兩個互斥（不重疊）的區間內所發生的事件的數目是互相獨立的隨機變數。

在區間 $[t, t + τ]$ 內發生的事件的數目的機率分佈為：

$P [(N(t+ \tau) - N(t)) = k] = \frac{e^{-\lambda \tau} (\lambda \tau)^k}{k!} \qquad k= 0,1,\ldots$

其中λ是一個正數，是固定的參數，通常稱為抵達率（arrival rate）或強度（intensity）。所以，如果給定在時間區間 $[t, t + τ]$ 之中事件發生的數目，則隨機變數 $N (t + τ) - N (t)$ 呈現泊松分布，其參數為 $λτ$ 。

更一般地來說，一個泊松過程是在每個有界的時間區間或在某個空間（例如：一個歐氏平面或三維的歐氏空間）中的每一個有界的區域，賦予一個隨機的事件數，使得

在一個時間區間或空間區域內的事件數，和另一個互斥（不重疊）的時間區間或空間區域內的事件數，這兩個隨機變數是獨立的。

在每一個時間區間或空間區域內的事件數是一個隨機變數，遵循泊松分布。（技術上而言，更精確地來說，每一個具有有限測度的集合，都被賦予一個泊松分布的隨機變數。）

-{zh-cn:泊松;zh-tw:卜瓦松;zh-hk:泊松}-過程是Lévy過程（Lévy process）中最有名的過程之一。時間齊次的-{zh-cn:泊松;zh-tw:卜瓦松;zh-hk:泊松}-過程也是時間齊次的連續時間Markov過程的例子。一個時間齊次、一維的-{zh-cn:泊松;zh-tw:卜瓦松;zh-hk:泊松}-過程是一個純出生過程，是一個出生-死亡過程的最簡單例子。

[编辑] 例子

After a short shower on a tiled pavement ( tiles of equal area), the total number of raindrops fallen on each tile will likely follow a Poisson distribution.

The number of telephone calls arriving at a switchboard during any specified time interval may have a Poisson distribution, and the number of calls arriving during one time interval may be statistically independent of the number of calls arriving during any other non-overlapping time interval. This is a one-dimensional Poisson process. In simple models, one may assume a constant average rate of arrival, e.g., λ = 12.3 calls per minute. In that case, the expected value of the number of calls in any time interval is that rate times the amount of time, λt. In messier and more realistic problems, one uses a non-constant rate function λ(t). In that case, the expected value of the number of calls between time a and time b is

$\int_a^b \lambda(t)\,dt.$

The number of photons hitting a photodetector during a specified time interval may follow a Poisson distribution.

If a CCD is exposed to the blue sky for a short time, the number of photons hitting each pixel may follow a Poisson distribution.

The number of bombs falling on a specified area of London in the early days of the Second World War may be a random variable with a Poisson distribution, and the number of bombs falling on two areas of the city that do not overlap may be statistically independent. The number of bombs observed to have fallen within an area A is a 2-dimensional Poisson process over the space defined by the area A.

Astronomers may treat the number of stars in a given volume of space as a random variable with a Poisson distribution, and the numbers of stars in any two or more non-overlapping regions as statistically independent. The number of stars observed within some volume V is a 3-dimensional Poisson process over the space defined by the volume V.

[编辑] 1-dimensional Poisson 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 distribution of the number N(t) of arrivals before time t follows a Poisson distribution with expected value λt.

Let X_t be the number of arrivals before time t. Let T_x 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 X_t has a discrete probability distribution -- a Poisson distribution -- and the random variable T_x 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 [ X_t < x ] occurs if and only if the event [ T_x > t ]. Consequently the probabilities of these events are the same:

P (X t < x) = P (T x > t).

This fact plus knowledge of the Poisson distribution enables us to find the probability distribution of these continuous random variables. In the case where 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

P (T 1 > t) = P (X t = 0) = e - λ t .

Consequently, the waiting time until the first arrival has an 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

$P(T_1>t+s \mid T_1>t)=P(T_1>s).$

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 > 30 are independent. To summarize: "memorylessness" of the probability distribution of the waiting time T₁ until the first arrival means

$\mathrm{(Right)}\ P(T_1>40 \mid T_1>30)=P(T_1>10).$

It does not mean

$\mathrm{(Wrong)}\ P(T_1>40 \mid T_1>30)=P(T_1>40).$

(That would be independence. These two events are not independent.)

An example: imagine a bus line where the departure time of buses is not scheduled at regular intervals, but is a Poisson process, so that you get an average of 1 bus/hour (100 buses start during a 100 hours interval, each one chooses the departure moment randomly according to a uniform distribution in the interval 0 - 100). Whenever you get to any bus stop, the average time you have to wait for the next bus is 1 hour. If you begin waiting just after a bus as passed, in average you'll have to wait 1 hour. If when you get to the bus stop there is a long queue, because no bus passed in the last five hours, the average time you'll have to wait is still 1 hour. So the average time you have to wait is independent of at which moment you arrive to the bus stop. But the two events "I'll have to wait more than 3 hours" and "I'll have to wait more than 4 hours" are NOT independent: having to wait more than four hours total is much more likely for people who have already spent three hours waiting (37%), than for people who have just arrived now (2%).

[编辑] Characterization of Poisson processes

In its most general form, the only two conditions for a 1-dimensional process to be a (not necessarily homogeneous) Poisson process are:

Orderliness: which roughly means

$\lim_{\Delta t\to 0} P(X_{t+\Delta t} - X_t > 1 \mid X_{t+\Delta t} - X_t \geq 1)=0$

which implies that arrivals don't occur simultaneously (but is actually a stronger statement). Simultaneous arrivals occur in some compound Poisson processes.

Memorylessness (also called evolution without aftereffects): the number of arrivals occurring in any bounded interval of time after time t is independent of the number of arrivals occurring before time t.

These seemingly unrestrictive conditions actually impose a great deal of structure in the Poisson process. In particular, they imply independent exponential (memoryless) interarrival times (with parameter λ for homogeneous processes). Because the interarrival times are exponentially distributed with rate $λ$ , the time between the 4th and 9th arrival (for instance) is distributed as the sum of exponential random variables (i.e. 5th order gamma distribution). Also, these conditions imply that the probability distribution of the number of events in the interval [a,b), which is also written as X_b − X_a is Poisson-distributed, (with parameter λ(b − a) for homogeneous processes).

This is a sample one-dimensional homogeneous Poisson process, X_t; not to be confused with a density or distribution function.