# 数学分析代写 Mathematical Analysis代考

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## 数学分析代写Mathematical Analysis

### 序列和限制Sequences and limits代写

• Real analysis实分析
• Complex analysis复分析

## 数学分析的历史

Mathematical analysis was formally developed during the scientific revolution of the 17th century, but many of its ideas can be traced back to early mathematicians. The earliest results of analysis are implied in early ancient Greek mathematics. For example, an infinite geometric sum is implied in Zeno’s dichotomous paradox. (Strictly speaking, the paradox aims to deny the existence of infinite sums.) Later, Greek mathematicians such as Eudoxus and Archimedes used the concepts of limit and convergence more explicitly but informally when they used the exhaustive method to calculate the area and volume of regions and entities. The explicit use of infinitesimals appears in Archimedes’ Method of Mechanical Theorems, a work rediscovered in the 20th century. In Asia, Chinese mathematician Liu Hui used the exhaustive method to find the area of a circle in the 3rd century CE. Jain literature shows that Indians had learned formulas for summing arithmetic and geometric series as early as the 4th century BCE. In Indian mathematics, particular examples of arithmetic series are found implicitly in the Vedic literature as early as 2000 BCE.

## 数学分析相关课后作业代写

Parameters with Order Restrictions. Let $X_1, \ldots, X_n$ be indpendent random variables with $$X_i \sim P_{\theta_i}, \text { for } i=1, \ldots, n$$ (a). For $P_\theta=N(\theta, 1)$, determine the maximum likelihood estimate of $$\left(\theta_1, \ldots, \theta_n\right)$$ when there are no restrictions on the $\theta_i$.

The likelihood function for the independent normal distribution is given by

$L\left(\theta_1, \ldots, \theta_n \mid x_1, \ldots, x_n\right)=\prod_{i=1}^n \frac{1}{\sqrt{2 \pi}} e^{-\frac{1}{2}\left(x_i-\theta_i\right)^2}$

The log-likelihood function is then

$\ell\left(\theta_1, \ldots, \theta_n \mid x_1, \ldots, x_n\right)=-\frac{n}{2} \log (2 \pi)-\sum_{i=1}^n \frac{1}{2}\left(x_i-\theta_i\right)^2$

To find the maximum likelihood estimates (MLEs), we differentiate the log-likelihood function with respect to each parameter and set the resulting equations equal to zero. Specifically,

$\frac{\partial \ell}{\partial \theta_i}=\sum_{j=1}^n\left(x_j-\theta_j\right) \frac{\partial \theta_j}{\partial \theta_i}=\theta_i-x_i=0 \quad$ for $i=1, \ldots, n$

Therefore, the MLE for $\theta_i$ is simply $\hat{\theta}_i = x_i$ for $i=1,\ldots,n$. This is intuitive since the normal distribution is symmetric and the maximum likelihood estimator for the mean is the sample mean.

Note that there are no restrictions on the $\theta_i$, so we don’t need to worry about any constraints on the estimates.