If $A$ is a subset of $R$ and $X$ is a random variable. I have two variables $X_1$ and $X_2$. $I$ being $1$ if $X$ in subset $A$ and $0$ if not in $A$. Let $U$~$U(0;1)$ and determine if this pair is independent. Verify your claim using simulation in Matlab.
$$ X_1 = I_U \epsilon\left[\left.0,\frac{1}{3}\right.\right), X_2 = I_U\epsilon\left[\left.\frac{1}{3},\frac{2}{3}\right.\right)$$
I usually show my work done, but I cannot find how to determine if these are independent. My question: Please, can someone explain how to show if this pair is independent? From there, then I can attempt how to verify using Matlab.
Intuitively, if we know that $X_1$ is one, then we know that $X_2$ is zero, so knowing $X_1$ means we know something about $X_2$, which suggests that they are not independent.
Explictly:
Let $A=\{1\}$. Compute $P[X_1 \in A], P[X_2 \in A], P[X_1 \in A, X_2 \in A]$.
Is $P[X_1 \in A, X_2 \in A] = P[X_1 \in A] P[X_2 \in A]$?