The indicator function (random variable) for a probability event $A \subset \Omega$ is given by

$ \mathbf{1}_A(x) =\begin{cases} 1 & \text{if }x \in A \\ 0 & \text{if }x \notin A. \end{cases}$

Consider $N$ dependent events $A_1,A_2,\cdots,A_N \subset \Omega.$ Now, we want to evaluate the probability $ \Pr \{A_1 \leq a,A_2 \leq a,\cdots,A_N \leq a\}$, which can be written in terms of the indicator functions as

$\Pr \{A_1 \leq a,A_2 \leq a,\cdots,A_N \leq a\} = E \left[ \mathbf{1}_{A_1 \leq a,A_2 \leq a,\cdots,A_N \leq a}\right].$ How to arrive at the next step in terms of covariance of the indicator random variables?

Example: Consider the 2 events case. Then,

$$\begin{align} \Pr \{A_1 \leq a,A_2 \leq a\} &= E \left[ \mathbf{1}_{A_1 \leq a,A_2 \leq a}\right]\\ &=E \left[ \mathbf{1}_{A_1 \leq a} \mathbf{1}_{A_2 \leq a}\right]\\ &=E \left[ \mathbf{1}_{A_1 \leq a} \right] E \left[ \mathbf{1}_{A_2 \leq a}\right]+\operatorname{Cov}\left(\mathbf{1}_{A_1 \leq a},\mathbf{1}_{A_2 \leq a} \right)\end{align}$$

The indicator function for a probability event $A \subset \Omega$ is given by

$ \mathbf{1}_A(x) =\begin{cases} 1 & \text{if }x \in A \\ 0 & \text{if }x \notin A. \end{cases}$

Consider $N$ dependent events $A_1 \leq a,A_2 \leq a,\cdots,A_N \leq a \subset \Omega.$ Now, we want to evaluate the probability $ \Pr \{A_1 \leq a,A_2 \leq a,\cdots,A_N \leq a\}$, which can be written in terms of the indicator functions as

$\Pr \{A_1 \leq a,A_2 \leq a,\cdots,A_N \leq a\} = E \left[ \mathbf{1}_{A_1 \leq a,A_2 \leq a,\cdots,A_N \leq a}\right].$ How to arrive at the next step in terms of covariance of the indicator random variables?

Example: Consider the 2 events case. Then,

$$\begin{align} \Pr \{A_1 \leq a,A_2 \leq a\} &= E \left[ \mathbf{1}_{A_1 \leq a,A_2 \leq a}\right]\\ &=E \left[ \mathbf{1}_{A_1 \leq a} \mathbf{1}_{A_2 \leq a}\right]\\ &=E \left[ \mathbf{1}_{A_1 \leq a} \right] E \left[ \mathbf{1}_{A_2 \leq a}\right]+\operatorname{Cov}\left(\mathbf{1}_{A_1 \leq a},\mathbf{1}_{A_2 \leq a} \right)\end{align}$$

The indicator function (random variable) for a probability event $A \subset \Omega$ is given by

$ \mathbf{1}_A =\left\{ \begin{array}{cc} 1 & if~~x \in A \\ 0 & if~~x \notin A \end{array} \right.$

Consider $N$ dependent events $A_1,A_2,\cdots,A_N \subset \Omega.$ Now, we want to evaluate the probability $ \Pr \{A_1 \leq a,A_2 \leq a,\cdots,A_N \leq a\}$, which can be written in terms of the indicator functions as

$ \Pr \{A_1 \leq a,A_2 \leq a,\cdots,A_N \leq a\} = E \left[ \mathbf{1}_{A_1 \leq a,A_2 \leq a,\cdots,A_N \leq a})\right].$ How to arrive at the next step in terms of covariance of the indicator random variables?

Example: Consider the 2 events case. Then,

$ \Pr \{A_1 \leq a,A_2 \leq a\} = E \left[ \mathbf{1}_{A_1 \leq a,A_2 \leq a}\right] =E \left[ \mathbf{1}_{A_1 \leq a} \mathbf{1}_{A_2 \leq a}\right]=E \left[ \mathbf{1}_{A_1 \leq a} \right] E \left[ \mathbf{1}_{A_2 \leq a}\right]+Cov\left(\mathbf{1}_{A_1 \leq a},\mathbf{1}_{A_2 \leq a} \right)$

The indicator function (random variable) for a probability event $A \subset \Omega$ is given by

$ \mathbf{1}_A(x) =\begin{cases} 1 & \text{if }x \in A \\ 0 & \text{if }x \notin A. \end{cases}$

Consider $N$ dependent events $A_1,A_2,\cdots,A_N \subset \Omega.$ Now, we want to evaluate the probability $ \Pr \{A_1 \leq a,A_2 \leq a,\cdots,A_N \leq a\}$, which can be written in terms of the indicator functions as

$\Pr \{A_1 \leq a,A_2 \leq a,\cdots,A_N \leq a\} = E \left[ \mathbf{1}_{A_1 \leq a,A_2 \leq a,\cdots,A_N \leq a}\right].$ How to arrive at the next step in terms of covariance of the indicator random variables?

Example: Consider the 2 events case. Then,

$$\begin{align} \Pr \{A_1 \leq a,A_2 \leq a\} &= E \left[ \mathbf{1}_{A_1 \leq a,A_2 \leq a}\right]\\ &=E \left[ \mathbf{1}_{A_1 \leq a} \mathbf{1}_{A_2 \leq a}\right]\\ &=E \left[ \mathbf{1}_{A_1 \leq a} \right] E \left[ \mathbf{1}_{A_2 \leq a}\right]+\operatorname{Cov}\left(\mathbf{1}_{A_1 \leq a},\mathbf{1}_{A_2 \leq a} \right)\end{align}$$