Let \(X_1\) and \(X_2\) be two independent tosses of a fair coin. Let \(Y\) be the  random variable  equal to 1 if exactly one of those coin tosses resulted in heads, and 0 otherwise. For simplicity, let 1 denote heads and 0 tails.

1. Write down the joint probability mass function for \((X_1,X_2,Y)\).
2. Show that \(X_1\) and \(X_2\) are independent.
3. Show that \(X_1\) and \(Y\) are independent.
4. Show that \(X_2\) and \(Y\) are independent.
5. The three variables are mutually independent if  for any \(x_1 \in Range(X_1), x_2\in Range(X_2), y \in Range(Y)\) one has
   \[ \mathbf{P}(X_1=x_1,X_2=x_2,Y=y) = \mathbf{P}(X_1=x_1) \mathbf{P}(X_2=x_2) \mathbf{P}(Y=y) \]
   Show that \(X_1,X_2,Y\) are not mutually independent.