7. The random variable \(X\) is defined as
$$X = 4 Y - 3 W$$
where \(Y \sim \mathrm {~N} \left( 40,3 ^ { 2 } \right) , W \sim \mathrm {~N} \left( 50,2 ^ { 2 } \right)\) and \(Y\) and \(W\) are independent.
- Find \(\mathrm { P } ( X > 25 )\)
The random variables \(Y _ { 1 } , Y _ { 2 }\) and \(Y _ { 3 }\) are independent and each has the same distribution as \(Y\). The random variable \(A\) is defined as
$$A = \sum _ { i = 1 } ^ { 3 } Y _ { i }$$
The random variable \(C\) is such that \(C \sim \mathrm {~N} \left( 115 , \sigma ^ { 2 } \right)\)
Given that \(\mathrm { P } ( A - C < 0 ) = 0.2\) and that \(A\) and \(C\) are independent,
- find the variance of \(C\).