6 In each round of a quiz a contestant can answer up to three questions. Each correct answer scores 1 point and allows the contestant to go on to the next question. A wrong answer scores 0 points and the contestant is allowed no further question in that round. If all 3 questions are answered correctly 1 bonus point is scored, making a total score of 4 for the round. For a certain contestant, \(A\), the probability of giving a correct answer is \(\frac { 3 } { 4 }\), independently of any other question. The random variable \(X _ { r }\) is the number of points scored by \(A\) during the \(r ^ { \text {th } }\) round.

1. Find the probability generating function of \(X _ { r }\).
2. Use the probability generating function found in part (i) to find the mean and variance of \(X _ { r }\).
3. Write down an expression for the probability generating function of \(X _ { 1 } + X _ { 2 }\) and find the probability that \(A\) has a total score of 4 at the end of two rounds.