- \(\mathrm { E } ( a X + b Y + c ) = a \mathrm { E } ( X ) + b \mathrm { E } ( Y ) + c\),
- if \(X\) and \(Y\) are independent then \(\operatorname { Var } ( a X + b Y + c ) = a ^ { 2 } \operatorname { Var } ( X ) + b ^ { 2 } \operatorname { Var } ( Y )\).
\section*{Discrete distributions}
\(X\) is a random variable taking values \(x _ { i }\) in a discrete distribution with \(\mathrm { P } \left( X = x _ { i } \right) = p _ { i }\)
Expectation: \(\mu = \mathrm { E } ( X ) = \sum x _ { i } p _ { i }\)
Variance: \(\sigma ^ { 2 } = \operatorname { Var } ( X ) = \sum \left( x _ { i } - \mu \right) ^ { 2 } p _ { i } = \sum x _ { i } ^ { 2 } p _ { i } - \mu ^ { 2 }\)
Greg and Nilaya play a game with these dice.
Greg throws the black die and Nilaya throws the white die. Greg wins the game if he scores at least two more than Nilaya, otherwise Greg loses.
The probability of Greg winning the game is \(\frac { 1 } { 6 }\)
(b) Find the value of \(a\) and the value of \(b\)
Show your working clearly.
The random variable \(X = 2 W - 5\)
Given that \(\mathrm { E } ( X ) = 2.6\)
(c) find the exact value of \(\operatorname { Var } ( X )\)
END OF EXAMINATION