### 物理代写|电动力学代写electromagnetism代考|ELEC3104

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 物理代写|电动力学代写electromagnetism代考|Stochastic Integration

The idea or purpose of stochastic integration is to define a random variable $\mathcal{S}{t}=$ $$\int{0}^{t} \mathcal{Z}(s) d \mathcal{X}(s), \text { or } \int_{0}^{t} f(\mathcal{X}(s)) d \mathcal{X}(s)$$
where $S_{t}$ is a random or unpredictable quantity, depending in a particular manner on unpredictable entities $\mathcal{X}$ and $\mathcal{Z}$; and where
$$\mathcal{X},=(\mathcal{X}(s): 0<s \leq t), \quad \mathcal{Z},=(\mathcal{Z}(s): 0<s \leq t)$$
are stochastic processes and $\mathcal{S}_{t}$ depends on time $t$. In textbooks, the integrand is usually presented as $f(s)$, but $\mathcal{Z}(s)$ is used here in order to emphasise that the integrand is intended to be random.

The integrand $\mathcal{Z}(s)$ (or, when appropriate, $f(\mathcal{X}(s))$ ) is to be regarded as a measurable function-as is $\mathcal{X}(s)$ – with respect to a probability space $(\Omega, \mathcal{A}, P)$.
If $\mathcal{Z}(s)$ is a deterministic or non-random function $g(s)$ of $s$, its value at time $s$ is a definite (non-random) number which, whenever necessary, can be regarded as a degenerate random variable. If $\mathcal{Z}(s)$ is the same random variable for each $s$ in $t_{j-1} \leq s<t_{j}$, each $j$, then the process $\mathcal{Z}$ is a step function. (In textbooks, the term elementary function is often applied to this.)

The most important kind of stochastic integral is where $\mathcal{X},=(\mathcal{X}(s))_{0<s \leq t}$, is standard Brownian motion, and this particular case (called the Itô integral) is outlined here. The main steps are as follows.

I1 Suppose the integrand $\mathcal{Z}(s)$ is a step function, with constant random variable value $\mathcal{Z}(s)=\mathcal{Z}{j-1}$ for $t{j-1} \leq s<t_{j}, 0=t_{0}<t_{1}<\cdots<t_{n}=t$. Then define
$$\mathcal{S}{t}=\int{0}^{t} \mathcal{Z}(s) d \mathcal{X}(s):=\sum_{j=1}^{n} \mathcal{Z}{j-1}\left(\mathcal{X}\left(t{j}\right)-\mathcal{X}\left(t_{j-1}\right)\right)$$
In this case (that is, $\mathcal{Z}(s)$ a step function), the Itô isometry holds for expected values:
$$\mathbf{E}\left(\mathcal{S}{t}^{2}\right)=\mathbf{E}\left(\int{0}^{t}(\mathcal{Z}(s))^{2} d s\right)$$

I2 Suppose the process $\mathcal{Z}(s)$ (not necessarily a step function) satisfies
$$\mathrm{E}\left(\int_{0}^{t}(\mathcal{Z}(s))^{2} d s\right)<\infty$$
Then there exists a sequence of step functions (processes) $\left{\mathcal{Z}^{(p)}(s)\right}, p=$ $1,2,3, \ldots$ such that
$$\lim {p \rightarrow \infty} \mathrm{E}\left(\int{0}^{t}\left|\mathcal{Z}{j}^{(p)}(s)-\mathcal{Z}(s)\right|^{2} d s\right)=0$$ I3 For such $\mathcal{Z}(s)$, define its stochastic integral $\mathcal{S}{t}$ with respect to the process $\mathcal{X}(s)$ as
\begin{aligned} \mathcal{S}{t}=\int{0}^{t} \mathcal{Z}(s) d \mathcal{X}(s) &:=\lim {p \rightarrow \infty} \mathcal{S}{t}^{(p)}=\lim {p \rightarrow \infty} \int{0}^{t} \mathcal{Z}^{(p)}(s) d \mathcal{X}(s) \ &=\lim {p \rightarrow \infty} \sum{j_{p}=1}^{n_{p}} \mathcal{Z}{j{p}}^{(p)}\left(\mathcal{X}\left(t_{j_{p}}\right)-\mathcal{X}\left(t_{j_{p}-1}\right)\right) \end{aligned}
I4 If $\mathcal{X}$ is Brownian motion the latter limit exists.

## 物理代写|电动力学代写electromagnetism代考|Random Variation

The previous chapter makes reference to random variables as functions which are measurable with respect to some probability domain. This conception of random variation is quite technical, and the aim of this chapter is to illuminate it by focussing on some fundamental features.

In broad practical terms, random variation is present when unpredictable outcomes can, in advance of actual occurrence, be estimated to within some margin of error. For instance, if a coin is tossed we can usually predict that heads is an outcome which is no more or no less likely than tails. So if an experiment consists of ten throws of the coin, it is no surprise if the coin falls heads-up on, let us say, between four and six occasions. This is an estimated outcome of the experiment, with estimated margin of error.

In fact, with a little knowledge of binomial probability distributions, we can predict that there is approximately 40 per cent chance that heads will be thrown on four, five or six occasions out of the ten throws. So if a ten-throw trial is repeated one hundred times, the outcome should be four, five, or six heads for approximately four hundred of the one thousand throws.

Such knowledge enables us to estimate good betting odds for placing a wager that a toss of the coin will produce this outcome. This is the “naive or realistic” view.

Can this fairly easily understandable scenario be expressed in the technical language of probability theory, as in Chapter 1 above? What is the probability space $(\Omega, \mathcal{A}, P)$ ? What is the $P$-measurable function which represents the random variable corresponding to a single toss of a coin?

The following remarks are intended to provide a link between the “naive or realistic” view, and the “sophisticated or mathematical” interpretation of this underlying reality.

The possible outcomes of an experiment consisting of a single throw of the coin are $\mathrm{H}$ (for heads) and $\mathrm{T}$ (for tails). Suppose a sample space $\Omega$ for this

experiment consists of the pair of numbers, 0 and 1 . Let $\mathcal{A}$ be the family of all subsets of $\Omega$ :
$$\Omega={0,1}, \quad \mathcal{A}={\varnothing,{0},{1},{0,1}} ;$$
and define a probability measure $P$ by
$$P(\varnothing)=0, \quad P({0})=\frac{1}{2}, \quad P({1})=\frac{1}{2}, \quad P({0,1})=P(\Omega)=1 \text {. }$$
Then, trivially, $\mathcal{A}$ is a $\sigma$-algebra of subsets of $\Omega$, and $P$ is, trivially, countably additive $^{1}$ on $\mathcal{A}$, so $(\Omega, \mathcal{A}, P)$ is a probability measure space.

The set of outcomes of a single throw of a coin is the set $V={H, T}$, and the family of subsets of $V$ is
$$\mathcal{V}={\varnothing,{\mathrm{H}},{\mathrm{T}},{\mathrm{H}, \mathrm{T}}}$$
and $(V, V)$ is a measurable space. Define the following function to represent the coin tossing experiment:
$$\mathcal{X}: \Omega \mapsto V, \quad \mathcal{X}(0)=\mathrm{H}, \quad \mathcal{X}(1)=\mathrm{T}$$

## 物理代写|电动力学代写electromagnetism代考|Probability and Riemann Sums

Elementary statistical calculation is often learned by performing exercises such as the following.

Example 4 A sample of 100 individuals is selected, their individual weights are measured, and the results are summarized in Table 2.2. Estimate the mean weight and standard deviation of the weights in the sample.

Sometimes calculation of the mean and standard deviation is done by setting out the workings as in Table 2.3. The observed weights of the sample members are grouped or classified in intervals $I$, and the proportion of weights in each interval $I$ is denoted by $F(I)$. A representative weight $x$ is chosen from each interval $I$. The function $f(x)$ is $x^{2}$ since, in this case, these values are needed in order to estimate the variance. Completing the calculation, the estimate of the arithmetic mean weight in the sample is
$$\sum x F(I)=44 \mathrm{~kg},$$
while the variance of the weights is approximately
$$\sum x^{2} F(I)-(44)^{2}=2580-1936=644 \mathrm{~kg}^{2} .$$
The latter calculation, involving $\sum x^{2} F(I)$, has the form $\sum f(x) F(I)$ with $f(x)=x^{2}$. The expressions $\sum x F(I)$ and $\sum f(x) F(I)$ have the form of Riemann sums, in which the interval of real numbers $[0,100]$ is partitioned by the intervals $I$, and where each $x$ is a representative data-value in the corresponding interval $I$. Thus the sums
$$\sum x F(I) \text { and } \sum f(x) F(I)$$
are approximations to the Stieltjes (or Riemann-Stieltjes) integrals
$$\int_{J} x d F \text { and } \int_{J} f(x) d F, \text { respectively; }$$
the domain of integration $[0,100]$ being denoted by $J$.

## 物理代写|电动力学代写electromagnetism代考|Stochastic Integration

\mathcal{X},=(\mathcal{X}(s): 0<s \leq t), \quad \mathcal{Z},=(\mathcal{Z}(s): 0<s \leq t )


I1 假设被积函数从(s)是阶跃函数，具有恒定的随机变量值从(s)=从j−1为了吨j−1≤s<吨j,0=吨0<吨1<⋯<吨n=吨. 然后定义

I2 假设过程从(s)（不一定是阶跃函数）满足

I4 如果X是布朗运动，存在后一个极限。

## 物理代写|电动力学代写electromagnetism代考|Random Variation

Ω=0,1,一个=∅,0,1,0,1;

X:Ω↦在,X(0)=H,X(1)=吨

## 物理代写|电动力学代写electromagnetism代考|Probability and Riemann Sums

∑XF(我)=44 ķG,

∑X2F(我)−(44)2=2580−1936=644 ķG2.

∑XF(我) 和 ∑F(X)F(我)

∫ĴXdF 和 ∫ĴF(X)dF, 分别;

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## MATLAB代写

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