Apply E(aX+b) or Var(aX+b) formulas directly

2. The probability of winning a certain game at a funfair is \(p\). Aman plays the game 5 times and Boaz plays the game 8 times. The independent random variables \(X\) and \(Y\) denote the number of wins for Aman and Boaz respectively.

1. Given that \(\mathrm { E } ( X Y ) = 6 \cdot 4\), calculate \(p\).
2. Find \(\operatorname { Var } ( X Y )\).

1. \(\mathrm { E } ( a X + b Y + c ) = a \mathrm { E } ( X ) + b \mathrm { E } ( Y ) + c\),
2. 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

1 The continuous random variable \(X\) has variance 9 The discrete random variable \(Y\) has standard deviation 2 and is independent of \(X\) Find \(\operatorname { Var } ( X + Y )\)
Circle your answer.
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3 The discrete random variable \(X\) has probability distribution Show that \(\mathrm { E } ( 5 X - 7 ) = 3.5\)

1 The discrete random variable \(X\) has \(\operatorname { Var } ( X ) = 5\)
Find \(\operatorname { Var } ( 4 X - 3 )\) Circle your answer.
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