- The random variable \(X\) is defined as
$$X = 4 A - 3 B$$
where \(A\) and \(B\) are independent and
$$A \sim \mathrm {~N} \left( 15,5 ^ { 2 } \right) \quad B \sim \mathrm {~N} \left( 10,4 ^ { 2 } \right)$$
- Find \(\mathrm { P } ( X < 40 )\)
The random variable \(C\) is such that \(C \sim \mathrm {~N} \left( 20 , \sigma ^ { 2 } \right)\)
The random variables \(C _ { 1 } , C _ { 2 }\) and \(C _ { 3 }\) are independent and each has the same distribution as \(C\)
The random variable \(D\) is defined as
$$D = \sum _ { i = 1 } ^ { 3 } C _ { i }$$
Given that \(\mathrm { P } ( A + B + D < 76 ) = 0.2420\) and that \(A , B\) and \(D\) are independent, - showing your working clearly, find the standard deviation of \(C\)