The independent random variables CCC and DDD represent the volumes, in millilitres, of coffee dispensed by 'Classic' and 'Deluxe' pods respectively, where
C∼N(45,1.22)andD∼N(60,1.62) C \sim \mathrm{N}(45, 1.2^2) \quad \text{and} \quad D \sim \mathrm{N}(60, 1.6^2) C∼N(45,1.22)andD∼N(60,1.62)The random variables C1,C2C_1, C_2C1,C2 are independent and each has the same distribution as CCC. The random variables D1,D2,D3D_1, D_2, D_3D1,D2,D3 are independent and each has the same distribution as DDD.
Given that the random variable MMM is defined as
M=C1+C2+D1+D2+D35 M = \frac{C_1 + C_2 + D_1 + D_2 + D_3}{5} M=5C1+C2+D1+D2+D3find P(M<52.5)P(M < 52.5)P(M<52.5).
The random variable VVV represents the volume dispensed by a 'Value' pod, such that V∼N(μ,3.52)V \sim \mathrm{N}(\mu, 3.5^2)V∼N(μ,3.52).
Given that P(V−C<8)=0.12P(V - C < 8) = 0.12P(V−C<8)=0.12 and that VVV and CCC are independent,
find the value of μ\muμ, giving your answer to 3 significant figures.
