The mass of two trace isotopes, UUU and VVV, found in soil samples can be modeled as independent normal random variables such that:
U∼N(62,52)andV∼N(28,42) U \sim \text{N}(62, 5^2) \quad \text{and} \quad V \sim \text{N}(28, 4^2) U∼N(62,52)andV∼N(28,42)A specific geological index DDD is calculated using the formula D=2U−3VD = 2U - 3VD=2U−3V.
Determine the probability that the index DDD is less than 30.
A third isotope SSS has a mass distributed as S∼N(45,σ2)S \sim \text{N}(45, \sigma^2)S∼N(45,σ2). A researcher takes one measurement of UUU, one of VVV, and five independent measurements of SSS, denoted S1,S2,S3,S4, and S5S_1, S_2, S_3, S_4, \text{ and } S_5S1,S2,S3,S4, and S5.
The total mass MMM is defined as M=U+V+∑i=15SiM = U + V + \sum_{i=1}^5 S_iM=U+V+∑i=15Si.
Given that P(M>350)=0.0143P(M > 350) = 0.0143P(M>350)=0.0143 and that U,V, and SiU, V, \text{ and } S_iU,V, and Si are all independent,
calculate the value of σ\sigmaσ, the standard deviation of SSS.
