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.
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.