4 The independent random variables \(X\) and \(Y\) have the distributions \(\operatorname { Po } ( 2 )\) and \(\operatorname { Po } ( 3 )\) respectively.
- Given that \(X + Y = 5\), find the probability that \(X = 1\) and \(Y = 4\).
- Given that \(\mathrm { P } ( X = r ) = \frac { 2 } { 3 } \mathrm { P } ( X = 0 )\), show that \(3 \times 2 ^ { r - 1 } = r\) ! and verify that \(r = 4\) satisfies this equation.