The independent random variables AAA and BBB are defined as:
A∼N(25,62)andB∼N(12,42) A \sim \text{N}(25, 6^2) \quad \text{and} \quad B \sim \text{N}(12, 4^2) A∼N(25,62)andB∼N(12,42)The random variable XXX is defined as X=3A−2BX = 3A - 2BX=3A−2B.
Find P(X>60)P(X > 60)P(X>60).
The random variable C∼N(30,σ2)C \sim \text{N}(30, \sigma^2)C∼N(30,σ2). The random variables C1,C2,C3, and C4C_1, C_2, C_3, \text{ and } C_4C1,C2,C3, and C4 are independent and each has the same distribution as CCC.
The random variable YYY is defined as Y=∑i=14CiY = \sum_{i=1}^4 C_iY=∑i=14Ci.
Given that P(A+B+Y<140)=0.0401P(A + B + Y < 140) = 0.0401P(A+B+Y<140)=0.0401 and that A,B, and YA, B, \text{ and } YA,B, and Y are independent,
find the value of σ\sigmaσ, the standard deviation of CCC.
