1 The random variables \(X\) and \(Y\) are independent with \(X \sim \operatorname { Po } ( 5 )\) and \(Y \sim \operatorname { Po } ( 4 )\). \(S\) denotes the sum of 2 observations of \(X\) and 3 observations of \(Y\).
- Find \(\mathrm { E } ( S )\) and \(\operatorname { Var } ( S )\).
- The random variable \(T\) is defined by \(\frac { 1 } { 2 } X - \frac { 1 } { 4 } Y\). Show that \(\mathrm { E } ( T ) = \operatorname { Var } ( T )\).
- State which of \(S\) and \(T\) (if either) does not have a Poisson distribution, giving a reason for your answer.