The independent random variables L L\,L and S S\,S represent the masses, in kg, of large and small industrial components, such that
L∼N(6.5,0.82)andS∼N(2.1,0.52) L \sim \mathrm{N}(6.5, 0.8^2) \quad \text{and} \quad S \sim \mathrm{N}(2.1, 0.5^2) L∼N(6.5,0.82)andS∼N(2.1,0.52)The random variables L1,L2 L_1, L_2\,L1,L2 are independent and each has the same distribution as LLL. The random variables S1,S2,S3 S_1, S_2, S_3\,S1,S2,S3 are independent and each has the same distribution as SSS.
The random variable M M\,M is defined as the mean mass of a specific batch of these components:
M=L1+L2+S1+S2+S35 M = \frac{L_1 + L_2 + S_1 + S_2 + S_3}{5} M=5L1+L2+S1+S2+S3Find P(M>4.2)P(M > 4.2)P(M>4.2).
The random variable Q Q\,Q represents the mass of a quality-control weight, where Q∼N(μ,0.92)Q \sim \mathrm{N}(\mu, 0.9^2)Q∼N(μ,0.92).
Given that P(Q−L<−4.5)=0.22P(Q - L < -4.5) = 0.22P(Q−L<−4.5)=0.22 and that Q Q\,Q and L L\,L are independent,
find the value of μ\muμ, giving your answer to 3 significant figures.
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.