![]() To better understand this issue, let's look at an example. In some sense, both are implying that the number of arrivals in non-overlapping intervals are independent. Thinking of the Poisson process, the memoryless property of the interarrival times is consistent with the independent increment property of the Poisson distribution. Remember that a discrete random variable $X$ is said to be a Poisson random variable with parameter $\mu$, shown as $X \sim Poisson(\mu)$, if its range is $R_X=\a,x \geq 0. The study of the sequential hypothesis testing of Poisson processes goes back to Wald (Wald and Wolfowitz 22, Wald 21) more recent studies on it include Bayraktar et al. Poisson random variable: Here, we briefly review some properties of the Poisson random variable that we have discussed in the previous chapters. In practice, the Poisson process or its extensions have been used to model $-$ the number of car accidents at a site or in an area $-$ the location of users in a wireless network $-$ the requests for individual documents on a web server $-$ the outbreak of wars $-$ photons landing on a photodiode. Thus, we conclude that the Poisson process might be a good model for earthquakes. Other than this information, the timings of earthquakes seem to be completely random. For example, suppose that from historical data, we know that earthquakes occur in a certain area with a rate of $2$ per month. Let X (Xt)t0 be a Poisson process with intensity > 0 is either 0 or 1 by sequentially observing X, test. It is usually used in scenarios where we are counting the occurrences of certain events that appear to happen at a certain rate, but completely at random (without a certain structure). The Poisson process is one of the most widely-used counting processes.
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