2025 RI P2 Q5

A discrete random variable \(X\) has the following probability distribution:

\[\mathrm{P}(X=x) = k(x^2 + x) \text{ for}\hspace{0.5em} x = 1, 2, 3, 4, 5, \text{ where}\hspace{0.5em} k \text{ is a constant.}\]

1. Show that \(k = \frac{1}{70}\).[1]
2. Find the exact values of \(\mathrm{E}(X)\) and \(\mathrm{Var}(X)\).[3]
3. Two independent observations \(X_1\) and \(X_2\) are taken of \(X\). Find the probability that the difference between the two observations is at least \(3\).[3]