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.
390 exam-style questions on OCR (MEI) A Level Maths 2.4 Probability Distributions, covering 2.4.1 Recognise binomial situations, 2.4.2 Probability of success p, 2.4.3 Calculate binomial probabilities, 2.4.4 Mean of the binomial distribution, 2.4.5 Expected frequencies for binomial, 2.4.6 Probability functions and discrete random variables, 2.4.7 Numerical probabilities for a simple distribution, 2.4.8 Normal distribution as a model (A-level only), 2.4.9 Shape of the Normal curve (A-level only), 2.4.10 Linear transformation and standardising (A-level only), 2.4.11 Symmetry and inflection of Normal curve (A-level only), 2.4.12 Calculate probabilities from a Normal distribution (A-level only), 2.4.13 Model with probability distributions, and 2.4 Probability Distributions. Each one has a worked solution and a mark scheme showing where the marks go.