- The random variable \(A\) is defined as
$$A = 4 X - 3 Y$$
where \(X \sim \mathrm {~N} \left( 30,3 ^ { 2 } \right) , Y \sim \mathrm {~N} \left( 20,2 ^ { 2 } \right)\) and \(X\) and \(Y\) are independent.
Find
- \(\mathrm { E } ( A )\),
- \(\operatorname { Var } ( A )\).
The random variables \(Y _ { 1 } , Y _ { 2 } , Y _ { 3 }\) and \(Y _ { 4 }\) are independent and each has the same distribution as \(Y\). The random variable \(B\) is defined as
$$B = \sum _ { i = 1 } ^ { 4 } Y _ { i }$$
- Find \(\mathrm { P } ( B > A )\).