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数学代写|随机过程Stochastic Porcesses代考|STATS217 Nonhomogeneous Poisson processes

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数学代写|随机过程Stochastic Porcesses代考|Nonhomogeneous Poisson processes

In many applications, it is not realistic to assume that the average arrival rate of events of a counting process, ${N(t), t \geq 0}$, is constant. In practice, this rate generally depends on the variable $t$. For example, the average arrival rate of customers into a store is not the same during the entire day. Similarly, the average arrival rate of cars on a highway fluctuates between its maximum during rush hours and its minimum during slack hours. We will generalize the definition of a Poisson process to take this fact into account.

Definition 5.2.1. Let ${N(t), t \geq 0}$ be a counting process with independent increments. This process is called a nonhomogeneous (or nonstationary) Poisson process with intensity function $\lambda(t) \geq 0$, for $t \geq 0$, if $N(0)=0$ and
i) $P[N(t+\delta)-N(t)=1]=\lambda(t) \delta+o(\delta)$,
ii) $P[N(t+\delta)-N(t) \geq 2]=o(\delta)$.
Remark. The condition i) implies that the process ${N(t), t \geq 0}$ does not have stationary increments unless $\lambda(t) \equiv \lambda>0$. In this case, ${N(t), t \geq 0}$ becomes a homogeneous Poisson process, with rate $\lambda$.

As in the particular case when the average arrival rate of events is constant, we find that the number of events that occur in a given interval has a Poisson distribution.

数学代写|随机过程Stochastic Porcesses代考|Compound Poisson processes

Definition 5.3.1. Let $X_1, X_2, \ldots$ be independent and identically distributed random variables, and let $N$ be a random variable whose possible values are all positive integers and that is independent of the $X_k$ ‘s. The variable
$$S_N:=\sum_{k=1}^N X_k$$
is called a compound random variable.
We already gave the formulas for the mean and the variance of $S_N$ [see Eqs. (1.89) and (1.90)]. We will now prove these formulas.

Proposition 5.3.1. The mean and the variance of the random variable $S_N$ defined above are given, respectively, by
$$E\left[\sum_{k=1}^N X_k\right]=E[N] E\left[X_1\right]$$
and
$$V\left[\sum_{k=1}^N X_k\right]=E[N] V\left[X_1\right]+V[N]\left(E\left[X_1\right]\right)^2$$

数学代写|随机过程Stochastic Porcesses代考|Compound Poisson processes

$$S_N:=\sum_{k=1}^N X_k$$

$$E\left[\sum_{k=1}^N X_k\right]=E[N] E\left[X_1\right]$$

$$V\left[\sum_{k=1}^N X_k\right]=E[N] V\left[X_1\right]+V[N]\left(E\left[X_1\right]\right)^2$$

MATLAB代写

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