1 The random variable \(X\) has the distribution \(\mathrm { B } ( n , p )\).
- Show, from the definition, that the probability generating function of \(X\) is \(( q + p t ) ^ { n }\), where \(q = 1 - p\).
- The independent random variable \(Y\) has the distribution \(\mathrm { B } ( 2 n , p )\) and \(T = X + Y\). Use probability generating functions to determine the distribution of \(T\), giving its parameters.