6. The random variable \(W\) is defined as
$$W = 3 X - 4 Y$$
where \(X \sim \mathrm {~N} \left( 21,2 ^ { 2 } \right)\) and \(Y \sim \mathrm {~N} \left( 8.5 , \sigma ^ { 2 } \right)\) and \(X\) and \(Y\) are independent.
Given that \(\mathrm { P } ( W < 44 ) = 0.9\)
- find the value of \(\sigma\), giving your answer to 2 decimal places.
The random variables \(A _ { 1 } , A _ { 2 }\) and \(A _ { 3 }\) each have the same distribution as \(A\), where \(A \sim \mathrm {~N} \left( 28,5 ^ { 2 } \right)\)
The random variable \(B\) is defined as
$$B = 2 X + \sum _ { i = 1 } ^ { 3 } A _ { i }$$
where \(X , A _ { 1 } , A _ { 2 }\) and \(A _ { 3 }\) are independent.
- Find \(\mathrm { P } ( B \leqslant 145 \mid B > 120 )\)