Inner dependence of Independent random vectors If $X = (X_1,X_2)$ and $Y = (Y_1,Y_2)$ , $X$ and $Y$ are stochastically independent can $X_1$ and $Y_1$ be dependent?
Hint:
$$
P(X_1\in A,Y_1\in B)=P(X\in A\times\mathbb{R},Y\in B\times\mathbb{R}).
$$
Let us consider that all variables are continuous and let us consider probability density functions $f$ that are distinguished by their arguments.

The independence of $X$ and $Y$ means
$$f(X,Y)=f(X)(Y)$$
The right hand side can be written using the chain rule as
$$f(X)f(Y) = f(X_2,X_1)f(Y_2,Y_1)=f(X_2|X_1)f(X_1)f(Y_2|Y_1)f(Y_1)=f(X_2|X_1)f(Y_2|Y_1)f(X_1)f(Y_1)=f(X_2,Y_2|X_1,Y_1)f(X_1)f(Y_1)$$
where the last equality stems from the fact that $X_2$ and $Y_2$ do not occur in $f(X_1)f(Y_1)$ and $f(X_1)f(Y_1)$ define a probability density on $X_2,X_1$.

The left hand side can be also decomposed by the chain rule
$$f(X,Y)=f(X_2,Y_2|X_1,Y_1)f(X_1,Y_1)$$

From the equality of left and right sides we can see the independence of $X_1$ and $X_2$.