N1995 P2 Q6 [Modified]

An unbiased disc has a single dot marked on one side and two dots marked on the other side. The disc and an unbiased die are thrown and the random variable \(X\) is the sum of the numbers of dots showing on the disc and on the top of the die. Tabulate the probability distribution of \(X\). Write down \(\mathrm{E}(X)\) and hence show that \(\mathrm{Var}(X) = \frac{19}{6}\). Find \(\mathrm{P}(X \geq 4 | X \leq 7)\).

Two independent observations of \(X_1\) and \(X_2\) are taken of \(X\).

1. Find \(\mathrm{Var}(2X_1 - X_2 + \mathrm{Var}(2X_1 - X_2))\).
2. Find \(\mathrm{P}(X_1 - X_2 \geq 5)\). Hence find \(\mathrm{P}(|X_1 - X_2| \geq 5)\).