# 统计代写|随机过程代写stochastic process代考|STAT3021

## 统计代写|随机过程代写stochastic process代考|Exercises and Complements

Exercise 4.1 Assume that customers arrive at a store according to a renewal process and that, independently of the process and of each other, every customer makes a purchase with the same probability $p$. Show that those customers who make a purchase also form the “renewal points” of a renewal process. Let the time scale be expanded by a factor of $1 / p$. Find the interarrival distribution of the new process in terms of the original interarrival distribution.

Exercise 4.2 Let the Laplace transforms of $F(x)$ (the interarrival distribution) and $W_t(x)=P(Y(t) \leq x)$, the d.f. of Residual life time $Y(t)$ in a renewal process be $L_F(s)$ and $L_W(s)$ respectively. Show that
$L_w(s) e^{-s t}=\left[1-L_F(s)\right] \int_t^{\infty} e^{-s y} d H(y)$, where $H(y)$ is the renewal function.
Exercise 4.3 (a) If the random lifetime of an item has d.f. $F(x)$, what is the mean remaining life of an item of age $x$ ?
(b) Find the mean total life time of an item when the d.f. $F(x)=1-e^{-\lambda x}, x \geq 0, \lambda>0$. Hence show that the mean total life is approximately 2 times the mean life when the renewal process has been in operation for a long time.

Exercise 4.4 Let $\left{X_k, k \geq 1\right}$ be a renewal process, $X_k$ has distribution $F(t), N(t)$ is the renewal counting function, and $H(t)=E(N(t))$ is the renewal function.

1. Show that
(a) $\lim _{t \rightarrow \infty} H(t)=\infty$
(b) for every $t \geq 0,0 \leq H(t) \leq \frac{F(t)}{1-F(t)} \leq \frac{1}{1-F(t)}$
2. Let $H_2(t)=E\left(N^2(t)\right)$
Show that
(a) $H_2(t)=\sum_{n=1}^{\infty}(2 n-1) F_{(t)}^{(n)}$
(b) $H_2(t)=H(t)+2 \int_0^t H(t-u) d H(u)$

## 统计代写|随机过程代写stochastic process代考|Introduction and History

Branching process (popularly known as Galton-Watson process) dates back to 1874 when a mathematical model was formulated by Galton and Watson for the problem of “Extinction of families”. The model did not attract for a long time but during the last 40 years much attention has been devoted to it.

An important class of Markov processes with countably many states is the class of branching processes. Let us suppose that we are observing a physical system consisting of a finite number of particles either of the same type or of several different types. With the passage of time each particle can disappear or turn into a group of new particles, independently of other particles. Phenomena described by such a scheme are frequently encountered in natural science and technology, sociology and demography, for example, showers of cosmic rays, growth. of large organic cells, the development of biological populations and the spread of epidemics. All of these processes are characterized by the same property, namely, their development has a branching form. A precise definition of processes of this type within the frame-work of the theory of Markov process brings us to the concept of a branching process. In fact, the mathematical model of such kind of emperical processes lead to the idea of a branching process.

The theory of branching random processes is a rapidly developing area of general stochastic processes. A great number of papers and the books of Harris (1963), Jagers (1975), Athreya and Ney (1972), Assmussen and Hering (1983) and other books are some of the results of this growth and they show the development of the theory. The interest in this theory is connected with its applications to a wide spectrum of practical problems. They include the description of various biological populations, the investigation of the transformation processes of particles in nuclear reactors, the investigation of cascade processes, of chemical processes, problems of queueing theory, the theory of graphs and other problems.
Suppose we start with an initial set of objects (or individuals) which form the Oth generation of these objects called ancestors. The offsprings reproduced or the objects generated by the objects of the Oth generation are the “direct descendents” of the ancestors and are said to form the first generation; the objects generated by those of the first generation form the second generation, and so no.

# 随机过程代考

## 统计代写|随机过程代写stochastic process代考|Exercises and Complements

$L_w(s) e^{-s t}=\left[1-L_F(s)\right] \int_t^{\infty} e^{-s y} d H(y)$ ，在哪里 $H(y)$ 是更新函数。

(b) 当 $\operatorname{df} F(x)=1-e^{-\lambda x}, x \geq 0, \lambda>0$. 因此表明，当更新过程已运行很长时间时，平均总寿命约为 平均寿命的 2 倍。

1. 表明
(a) $\lim _{t \rightarrow \infty} H(t)=\infty$
(b) 对于每个 $t \geq 0,0 \leq H(t) \leq \frac{F(t)}{1-F(t)} \leq \frac{1}{1-F(t)}$
2. 让 $H_2(t)=E\left(N^2(t)\right)$
表明
(a) $H_2(t)=\sum_{n=1}^{\infty}(2 n-1) F_{(t)}^{(n)}$
(二) $H_2(t)=H(t)+2 \int_0^t H(t-u) d H(u)$

## 统计代写|随机过程代写stochastic process代考|Introduction and History

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