in my homework i have the following excercise:
May A and B be events. May there be a probability space $(\Omega, \mathcal{F}, P)$. Show that $E(1_A) = P(A)$. Show that A and B are independent exactly when $E(1_A \cdot 1_B) = E(1_A) \cdot E(1_B)$.
A random variable $1_A$ is $1_A: \Omega \rightarrow \{0, 1\}$.
My question for the the question in general: The definition of $E(X)$ is $\int^{\infty}_{-\infty} x \cdot f(x) dx$. Now i don´t have any f here. Or do i have to extract the f from the definition of $1_A$? Like $ f: \Omega \rightarrow \{0, 1\}$ and $x \rightarrow \{0, 1\}$? If yes, how do i go about this?
For the second part i can imagine, if i can show that $E(1_A) = P(A)$ i can insert this in the definition of normal independence which is $P(A\cap B) = P(A) \cdot P(B)$.
Would be cool if you could give me a tip how to get the f. Maybe i´m thinking too complicated.
1) If $\Omega$ is a probability space with probability measure $\mathbb{P}$, the expected value of $1_A$ is defined as the integral of $1_A$ with respect to $\mathbb{P}$:
$E(1_A) = \int_\Omega 1_A(\omega) d\mathbb{P}(\omega) = \int_A d\mathbb{P}$ = P(A)
2) Using this:
($\to$) Suppose that A and B are independent. Then $E(1_A 1_B)$ = $E(1_{A\cap B}$) = $P(A\cap B)$ = $P(A)P(B)$ = $E(1_A)E(1_B)$
($\leftarrow$) We know that $E(1_A 1_B)$ = $E(1_A)E(1_B)$ holds. Then $P(A\cap B)$ = $E(1_{A\cap B})$ = $E(1_A 1_B)$ = $E(1_A)E(1_B)$ = $P(A)P(B)$.