# 统计代写| Geometric and Negative Binomial stat代写

## 统计代考

4.4 Indicator r.v.s and the fundamental bridge
This section is devoted to indicator random variables, which we already encountered in the previous chapter but will treat in much greater detail here. In particular, we will show that indicator r.v.s are an extremely useful tool for calculating expected values.
Recall from the previous chapter that the indicator r.v. $I_{A}$ (or $I(A)$ ) for an event $A$ is defined to be 1 if $A$ occurs and 0 otherwise. So $I_{A}$ is a Bernoulli random variable, where success is defined as ” $A$ occurs” and failure is defined as ” $A$ does not occur”.
Some useful properties of indicator r.v.s are summarized below.
Theorem 4.4.1 (Indicator r.v. properties). Let $A$ and $B$ be events. Then the following properties hold

1. $\left(I_{A}\right)^{k}=I_{A}$ for any positive integer $k$.
2. $I_{A^{\prime}}=1-I_{A}$.
3. $I_{A \cap B}=I_{A} I_{B}$.
4. $I_{A \cup B}=I_{A}+I_{B}-I_{A} I_{B}$
Proof. Property 1 holds since $0^{k}=0$ and $1^{k}=1$ for any positive integer $k$. Property 2 holds since $1-I_{A}$ is 1 if $A$ does not occur and 0 if $A$ occurs. Property 3 holds since $I_{A} I_{B}$ is 1 if both $I_{A}$ and $I_{B}$ are 1 , and 0 otherwise. Property 4 holds since
$$I_{A \cup B}=1-I_{A \oplus \cap B^{}}=1-I_{A^{}} I_{B^{*}}=1-\left(1-I_{A}\right)\left(1-I_{B}\right)=I_{A}+I_{B}-I_{A} I_{B} .$$
Indicator r.v.s provide a link between probability and expectation; we call this fact the fundamental bridge.
Theorem 4.4.2 (Fundamental bridge between probability and expectation). There is a one-to-one correspondence between events and indicator r.v.s, and the probability of an event $A$ is the expected value of its indicator r.v. $I_{A}$ :
$$P(A)=E\left(I_{A}\right)$$
Proof. For any event $A$, we have an indicator r.v. $I_{A}$. This is a one-to-one correspondence since $A$ uniquely determines $I_{A}$ and vice versa (to get from $I_{A}$ back to $A$, we can use the fact that $\left.A=\left{s \in S: I_{A}(s)=1\right}\right)$. Since $I_{A} \sim \operatorname{Bern}(p)$ with $A$, we can use the fact that $A=$ – $p=P(A)$, we have $E\left(I_{A}\right)=P(A)$.

The fundamental bridge connects events to their indicator r.v.s. and allows us to express any probability as an expectation. As an example, we give a short proof of inclusion-exelusion and a related inequality known as Boole’s inequality or Bonferroni’s inequality using indicator r.v.s.

## 统计代考

4.4 指标 r.v.s 和基本桥

1. 对于任何正整数 $k$，$\left(I_{A}\right)^{k}=I_{A}$。
2. $I_{A^{\prime}}=1-I_{A}$。
3. $I_{A \cap B}=I_{A} I_{B}$。
4. $I_{A \cup B}=I_{A}+I_{B}-I_{A} I_{B}$
证明。对于任何正整数 $k$，属性 1 成立，因为 $0^{k}=0$ 和 $1^{k}=1$。属性 2 成立，因为如果 $A$ 不出现，$1-I_{A}$ 为 1，如果 $A$ 出现，则为 0。属性 3 成立，因为如果 $I_{A}$ 和 $I_{B}$ 都为 1，则 $I_{A} I_{B}$ 为 1，否则为 0。财产 4 持有以来
$$I_{A \cup B}=1-I_{A \oplus \cap B^{}}=1-I_{A^{}} I_{B^{*}}=1-\left(1- I_{A}\right)\left(1-I_{B}\right)=I_{A}+I_{B}-I_{A} I_{B} 。$$
指标 r.v.s 提供概率和期望之间的联系；我们称这个事实为根本的桥梁。
定理 4.4.2（概率和期望之间的基本桥梁）。事件与指标 r.v.s 之间存在一一对应关系，事件 $A$ 的概率是其指标 r.v.的期望值。 $I_{A}$ ：
$$P(A)=E\left(I_{A}\right)$$
证明。对于任何事件 $A$，我们都有一个指标 r.v。 $I_{A}$。这是一一对应的，因为 $A$ 唯一确定 $I_{A}$，反之亦然（要从 $I_{A}$ 回到 $A$，我们可以使用 $\left. A=\left{s \in S: I_{A}(s)=1\right}\right)$。由于 $I_{A} \sim \operatorname{Bern}(p)$ 和 $A$，我们可以使用 $A=$ – $p=P(A)$ 的事实，我们有 $E\left(I_ {A}\right)=P(A)$。