### 数学代写|离散数学作业代写discrete mathematics代考|MATH300

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• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
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• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|离散数学作业代写discrete mathematics代考|Random variables

Let us now recall the definition of a generic random variable, and then the specific case of discrete random variables.

DEFINITION 1.9.-Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probabilizable space and $(E, \mathcal{E})$ be a measurable space. A random variable on the probability space $(\Omega, \mathcal{F}, \mathbb{P})$ taking values in the measurable space $(E, \mathcal{E})$, is any mapping $X: \Omega \longrightarrow E$ such that, for any $B$ in $\mathcal{E}, X^{-1}(B) \in \mathcal{F}$; in other words, $X: \Omega \longrightarrow E$ is a random variable if it is an $(\mathcal{F}, \mathcal{E})$-measurable mapping. We then write the event ” $\mathrm{X}$ belongs to $\mathrm{B}$ ” by
$$X^{-1}(B)={\omega \in \Omega ; X(\omega) \in B}=(X \in B) .$$
In the specific case where $E=\mathbb{R}$ and $\mathcal{E}=\mathcal{B}(\mathbb{R})$, the mapping $X$ is called a real random variable. If $E=\mathbb{R}^d$ with $d \geq 2$, and $\mathcal{E}=\mathcal{B}\left(\mathbb{R}^d\right)$, the mapping $X$ is said to be a real random vector.

EXAMPLE 1.12.- Let us return to the experiment where a six-sided die is rolled, where the set of possible outcomes is $\Omega={1,2,3,4,5,6}$, which is endowed with the uniform probability. Consider the following game:

• if the result is even, you win $10 €$;
• if the result is odd, you win $20 €$.
This game can be modeled using the random variable $X: \Omega \longmapsto{10,20}$, defined by:
$$X(\omega)=\left{\begin{array}{l} 10 \text { if } \omega \in{2,4,6} \ 20 \text { if } \omega \in{1,3,5} . \end{array}\right.$$
This mapping is a random variable, since for any $B \in \mathcal{P}({10,20})$, we have
(X \in B)=X^{-1}(B)=\left{\begin{aligned} {2,4,6} & \text { if } B={10} \ {1,3,5} & \text { if } B={20} \ \Omega \text { if } B={10,20} \ \emptyset \text { if } B=\emptyset . \end{aligned}\right.
and all these events are in $\mathcal{P}(\Omega)$.

## 数学代写|离散数学作业代写discrete mathematics代考|σ-algebra generated by a random variable

We now define the $\sigma$-algebra generated by a random variable. This concept is important for several reasons. For instance, it can make it possible to define the independence of random variables. It is also at the heart of the definition of conditional expectations; see Chapter 2.

PROPOSITION 1.6.- Let $X$ be a real random variable, defined on $(\Omega, \mathcal{F}, \mathbb{P})$ taking values in $(E, \mathcal{E})$. Then, $\mathcal{F}_X=X^{-1}(\mathcal{E})=\left{X^{-1}(A) ; A \in \mathcal{E}\right}$ is a sub- $\sigma$-algebra of $\mathcal{F}$ on $\Omega$. This is called the $\sigma$-algebra generated by the random variable $X$. It is written as $\sigma(X)$. It is the smallest $\sigma$-algebra on $\Omega$ that makes $X$ measurable:
$$\sigma(X)=X^{-1}(\mathcal{B}(\mathbb{R}))=\left{X^{-1}(B) ; B \in \mathcal{B}(\mathbb{R})\right}={(X \in B) ; B \in \mathcal{B}(\mathbb{R})} .$$
EXAMPLE 1.19.- Let $\mathcal{F}_0={\emptyset, \Omega}$ and $X=c \in \mathbb{R}$ be a constant. Then, for any Borel set $B$ in $\mathbb{R},(X \in B)$ has the value $\emptyset$ if $c \notin B$ and $\Omega$ if $c \in B$. Thus, the $\sigma$-algebra generated by $X$ is $\mathcal{F}_0$. Reciprocally, it can be demonstrated that the only $\mathcal{F}_0$-measurable random variables are the constants. Indeed, let $X$ be a $\mathcal{F}_0$-measurable random variable. Assume that it takes at least two different values, $x$ and $y$. It may be assumed that $y \geq x$ without loss of generality. Therefore, let $B=\left[x, \frac{x+y}{2}\right]$. We have that $(X \in B)$ is non-empty because $x \in B$ but is not $\Omega$ since $y \notin B$. Therefore, $X$ is not $\mathcal{F}_0$-measurable.

Proposition 1.7.- Let $X$ be a random variable on $(\Omega, \mathcal{F}, \mathbb{P})$ taking values in $(E, \mathcal{E})$ and let $\sigma(X)$ be the $\sigma$-algebra generated by $X$. Thus, a random variable $Y$ is $\sigma(X)$-measurable if and only if there exists a measurable function $f$ such that $Y=f(X)$.

This technical result will be useful in certain demonstrations further on in the text. In general, if it is known that $Y$ is $\sigma(X)$-measurable, we cannot (and do not need to) make explicit the function $f$. Reciprocally, if $Y$ can be written as a measurable function of $X$, it automatically follows that $Y$ is $\sigma(X)$-measurable.

EXAMPLE 1.20.- A die is rolled 2 times. This experiment is modeled by $\Omega={1,2,3,4,5,6}^2$ endowed with the $\sigma$-algebra of its subsets and the uniform distribution. Consider the mappings $X_1, X_2$ and $Y$ from $\Omega$ onto $\mathbb{R}$ defined by
\begin{aligned} &X_1\left(\omega_1, \omega_2\right)=\omega_1, \ &X_2\left(\omega_1, \omega_2\right)=\omega_2, \ &Y\left(\omega_1, \omega_2\right)=1_{{2,4,6}}\left(\omega_1\right), \end{aligned}
thus, $X_i$ is the result of the ith roll and $Y$ is the parity indicator of the first roll. Therefore, $Y=1_{{2,4,6}}\left(X_1\right)$; thus, $Y$ is $\sigma\left(X_1\right)$-measurable. On the other hand, $Y$ cannot be written as a function of $X_2$.

## 数学代写|离散数学作业代写discrete mathematics代考|Random variables

$X: \Omega \longrightarrow E$ 是一个随机变量，如果它是 $(\mathcal{F}, \mathcal{E})$-可测量的映射。然后我们编写事件”X $\mathrm{X}$ 属于 $\mathrm{B}^{\prime \prime}$ 经过
$$X^{-1}(B)=\omega \in \Omega ; X(\omega) \in B=(X \in B) .$$

• 如果结果是偶数, 你就赢了 $10 €$;
• 如果结果是奇数，你赢了 $20 €$. 这个游戏可以使用随机变量建模 $X: \Omega \longmapsto 10,20$, 定义为:
$\$ \$$X( lomega )=\backslash left { 10 if \omega \in 2,4,620 if \omega \in 1,3,5. 、正确的。 Thismappingisarandomvariable, since forany \ B \in \mathcal{P}(10,20) \$$, wehave
$(X \backslash$ in $B)=X \wedge{-1}(B)=\backslash l \operatorname{lt}{$
$2,4,6$ if $B=101,3,5 \quad$ if $B=20 \Omega$ if $B=10,20 \emptyset$ if $B=\emptyset$.
、正确的。


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