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# 数学代写| 随机过程代考|Prediction and conditional expectation

A population starts with one individual at time $n=0: Z_{0}=1$.

After one unit of time (at time $n=1$ ) the sole individual produces $Z_{1}$ identical clones of itself and dies. $Z_{1}$ is an $\mathbb{N}_{0}$-valued random variable.

(a) If $Z_{1}$ happens to be equal to 0 the population is dead and nothing happens at any future time $n \geq 2$.

(b) If $Z_{1}>0$, a unit of time later, each of $Z_{1}$ individuals gives birth to a random number of children and dies. The first one has $Z_{1,1}$ children, the second one $Z_{1,2}$ children, etc. The last, $Z_{1}^{\text {th }}$ one, gives birth to $Z_{1, Z_{1}}$ children. We assume that the distribution of the number of children is the same for each individual in every generation and independent of either the number of individuals in the generation and of the number of children the others have. This distribution, shared by all $Z_{n, i}$ and $Z_{1}$, is called the offspring distribution. The total number of individuals in the second generation is now
$$Z_{2}=\sum_{k=1}^{Z_{1}} Z_{1, k}$$
(c) The third, fourth, etc. generations are produced in the same way. If it ever happens that $Z_{n}=0$, for some $n$, then $Z_{m}=0$ for all $m \geq n$ – the population is extinct. Otherwise,
$$Z_{n+1}=\sum_{k=1}^{Z_{n}} Z_{n, k}$$

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## 数学代写| 随机过程代考|mathematicians

Probability theory is not merely an intellectual pursuit, but provides also a framework for estimation and prediction. Practical men and women of ten need to make decisions based on quantities which are not easily measurable, either because they lie in the future or because of some intrinsic inaccessibility; in doing so they usually make use of some current or feasible observation. Economic examples are commonplace (business trends, inflation rates, and so on); other examples include weather prediction, the climate in prehistoric times, the state of the core of a nuclear reactor, the cause of a disease in an individual or a population, or the paths of celestial bodies. This last problem has the distinction of being amongst the first to be tackled by mathematicians using a modern approach to probability.

## 数学代写| 随机过程代考|triangle inequalit

At its least complicated, a question of prediction or estimation involves an unknown or unobserved random variable $Y$, about which we are provided with the value of some (observable) random variable $X$. The problem is to deduce information about the value of $Y$ from a knowledge of the value of $X$. Thus we seek a function $h(X)$ which is (in some sense) close to $Y$; we write $\widehat{Y}=h(X)$ and call $\widehat{Y}$ an ‘estimator’ of $Y$. As we saw in Section 7.1, there are many different ways in which two random variables may be said to be close to one anotherpointwise, in $r$ th mean, in probability, and so on. A particular way of especial convenience is to work with the norm given by
$$|U|_{2}=\sqrt{\mathbb{E}\left(U^{2}\right)},$$
so that the distance between two random variables $U$ and $V$ is
$$|U-V|_{2}=\sqrt{\mathbb{E}\left{(U-V)^{2}\right}}$$

## 数学代写| 随机过程代考|TRIANGLE INEQUALIT

|ü|2=和(ü2),

|UV|_{2}=\sqrt{\mathbb{E}\left{(UV)^{2}\right}}|UV|_{2}=\sqrt{\mathbb{E}\left{(UV)^{2}\right}}

## Matlab代写

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