1 The marks obtained by a randomly chosen student in the two papers of an examination are denoted by the random variables \(X\) and \(Y\), where \(X \sim \mathrm {~N} ( 45,81 )\) and \(Y \sim \mathrm {~N} ( 33,63 )\). The student's overall mark for the examination, \(T\), is given by \(T = X + \lambda Y\), where the constant \(\lambda\) is chosen such that \(\mathrm { E } ( T ) = 100\).

1. Show that \(\lambda = \frac { 5 } { 3 }\).
2. Assuming that \(X\) and \(Y\) are independent, state the distribution of \(T\), giving the values of its parameters.
3. Comment on the assumption of independence.