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.
616 exam-style questions on Edexcel A Level Maths The Normal Distribution, covering 3.1 The Normal Distribution, 3.2 Finding Probabilities for Normal Distributions, 3.3 The Inverse Normal Distribution Function, 3.4 The Standard Normal Distribution, 3.5 Finding the mean and standard deviation, 3.6 Approximating a Binomial Distribution, and 3.7 Hypothesis Testing with the Normal Distribution. Each one has a worked solution and a mark scheme showing where the marks go.