5 Two discrete random variables \(X\) and \(Y\) have a joint probability distribution defined by $$\mathrm { P } ( X = x , Y = y ) = a ( x + y + 1 ) \quad \text { for } x = 0,1,2 \text { and } y = 0,1,2 ,$$ where \(a\) is a constant.

1. Show that \(a = \frac { 1 } { 27 }\).
2. Find \(\mathrm { E } ( X )\).
3. Find \(\operatorname { Cov } ( X , Y )\).
4. Are \(X\) and \(Y\) independent? Give a reason for your answer.
5. Find \(\mathrm { P } ( X = 1 \mid Y = 2 )\).