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# 统计代写|生存模型代考Survival Models代写|The Concept of Exposure

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## 统计代写|生存模型代考Survival Models代写|The Concept of Exposure

A survival model, expressed by the $\operatorname{SDF} S(x)$, is a probability distribution with all the properties that such distributions possess. The simple transformation of $\ell_x=\ell_0 \cdot S(x)$, where $\ell_0$ is a constant, expresses the same probability distribution. The function $\ell_x$ is the same function as is $S(x)$, except for the trivial difference that whereas $1 \geq S(x) \geq 0$, we now have $\ell_0 \geq \ell_x \geq 0$. Any function or result that can be derived from $S(x)$ can also be derived from $\ell_x$.
However, an advantage in using $\ell_x$, instead of $S(x)$, is the ability to interpret the values of $\ell_x$ as the survivors of an initial, closed, cohort of newborn lives of size of $\ell_0$. Successive values of $S(x)$ are probabilities, which are somewhat abstract, especially to non-mathematicians. But values of $\ell_x$ have a “real world” meaning, notwithstanding the fact that we are dealing with hypothetical situations.
In turn, the interpretive nature of $\ell_x$ allows for concrete (albeit hypothetical) interpretations of several functions derived from $\ell_x$. Of particular usefulness is the function $L_x$, defined by (3.37).
Recall, from (3.26), that ${ }s p_x \mu{x+s}$ is the PDF for death at age $x+s$, given alive at age $x$. If we multiply this PDF by $\ell_x$, which we interpret as the number of persons alive in a group at age $x$, we obtain $\ell_{x+s} \mu_{x+s}$, which is the rate of deaths occurring in the group at exact age $x+s$. In turn, $\ell_{x+s} \mu_{x+s} d s$ is the differential number of deaths occurring at exact age $x+s$. Then $s \cdot \ell_{x+s} \mu_{x+s} d s$ is the total number of years lived by those deaths after attaining age $x$. Finally, $\int_0^1 s \cdot \ell_{x+s} \mu_{x+s} d s$ gives the aggregate number of years lived, after age $x$, by all those who die between age $x$ and age $x+1$.
Most of the $\ell_x$ group, of course, survive to age $x+1, \ell_{x+1}$ being the number who do so. Each of these persons live one year from age $x$ to age $x+1$, so $\ell_{x+1}$ also represents the aggregate number of years lived, between ages $x$ and $x+1$, by those who survive to age $x+1$. Together,
$$\ell_{x+1}+\int_0^1 s \cdot \ell_{x+s} \mu_{x+s} d s$$
gives the aggregate number of years lived between ages $x$ and $x+1$ by the $\ell_x$ persons who comprised the group at age $x$.

## 统计代写|生存模型代考Survival Models代写|Relationship between ${ }_n q_x$ and ${ }_n m_x$

Familiarity with the concept of exposure allows us to develop a very useful formula which relates the functions ${ }n q_x$ and ${ }_n m_x$. Let us first explore this relationship with $n=1$. Since $q_x=\frac{d_x}{\ell_x}$, and $m_x=\frac{d_x}{L_x}$, then we are, in effect, exploring the relationship between $\ell_x$ and $L_x$. To do this, we first need to define a new function. From (3.40), we recall that $$\int_0^1 s \cdot \ell{x+s} \mu_{x+s} d s=\int_0^1 \ell_{x+s} d s-\ell_{x+1}=L_x-\ell_{x+1}$$
gives the aggregate number of life-years lived in $(x, x+1$ ] by those who die in that age interval, namely $d_x$. Then if (3.46) is divided by $d_x$, we obtain the average number of years lived in $(x, x+1]$ by those who die in that interval. It is clear that this average number is necessarily less than one, and could also be called the average fraction of $(x, x+1]$ lived through by those who die in that interval. We define this average fraction to be $f_x$, so that
$$f_x=\frac{L_x-\ell_{x+1}}{d_x}$$
Further, since $d_x=\ell_x-\ell_{x+1}$, then $\ell_{x+1}=\ell_x-d_x$, so we have
$$f_x \cdot d_x=L_x-\ell_x+d_x$$
or
$$L_x=\ell_x-\left(1-f_x\right) d_x$$
Then
$$m_x=\frac{d_x}{L_x}=\frac{d_x}{\ell_x-\left(1-f_x\right) d_x}=\frac{q_x}{1-\left(1-f_x\right) q_x} .$$
Alternatively,
$$\ell_x=L_x+\left(1-f_x\right) d_x$$
so
$$q_x=\frac{d_x}{\ell_x}=\frac{d_x}{L_x+\left(1-f_x\right) d_x}=\frac{m_x}{1+\left(1-f_x\right) m_x} .$$

# 生存模型代考

## 统计代写|生存模型代考Survival Models代写|The Concept of Exposure

$$\ell_{x+1}+\int_0^1 s \cdot \ell_{x+s} \mu_{x+s} d s$$

## 统计代写|生存模型代考Survival Models代写|Relationship between ${ }_n q_x$ and ${ }_n m_x$

$$f_x=\frac{L_x-\ell_{x+1}}{d_x}$$

$$f_x \cdot d_x=L_x-\ell_x+d_x$$

$$L_x=\ell_x-\left(1-f_x\right) d_x$$

$$m_x=\frac{d_x}{L_x}=\frac{d_x}{\ell_x-\left(1-f_x\right) d_x}=\frac{q_x}{1-\left(1-f_x\right) q_x} .$$

$$\ell_x=L_x+\left(1-f_x\right) d_x$$

$$q_x=\frac{d_x}{\ell_x}=\frac{d_x}{L_x+\left(1-f_x\right) d_x}=\frac{m_x}{1+\left(1-f_x\right) m_x} .$$

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。