Pre-U Pre-U 9795/2 2013 November — Question 4

Exam BoardPre-U
ModulePre-U 9795/2 (Pre-U Further Mathematics Paper 2)
Year2013
SessionNovember
TopicLinear combinations of normal random variables
TypeDifferent variables, one observation each
DifficultyChallenging +1.2 This question tests standard Further Maths content on linear combinations of normal variables. Part (i) requires a routine MGF proof (standard technique), part (ii) is immediate from (i), and parts (iii)-(iv) apply the theory to a straightforward context problem with conditional probability. While it requires multiple techniques and careful calculation, all steps follow established methods without requiring novel insight.
Spec2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation5.03a Continuous random variables: pdf and cdf5.04b Linear combinations: of normal distributions
4 It is given that \(X\) and \(Y\) are independent random variables with distributions \(\mathrm { N } \left( \mu _ { x } , \sigma _ { x } ^ { 2 } \right)\) and \(\mathrm { N } \left( \mu _ { y } , \sigma _ { y } ^ { 2 } \right)\) respectively, and that \(W\) is a random variable such that \(W = X + Y\).
  1. Use moment generating functions to show that the distribution of \(W\) is \(\mathrm { N } \left( \mu _ { x } + \mu _ { y } , \sigma _ { x } ^ { 2 } + \sigma _ { y } ^ { 2 } \right)\).
  2. State the distribution of \(X - Y\). The diameters of the central poles of one brand of rotary clothes lines are normally distributed with mean 3.75 cm and variance \(0.000125 \mathrm {~cm} ^ { 2 }\). The diameters of the cylindrical tubes, into which the central poles fit, are normally distributed with mean 3.85 cm and variance \(0.0001 \mathrm {~cm} ^ { 2 }\). Poles and tubes are chosen at random. The 'clearance' between a tube and a pole is the diameter of the tube minus the diameter of the pole.
  3. Find the probability that a pole and tube have a clearance between 0.08 cm and 0.13 cm .
  4. Given that a pole and tube have a clearance between 0.08 cm and 0.13 cm , find the probability that the clearance is between 0.11 cm and 0.125 cm .
4 It is given that $X$ and $Y$ are independent random variables with distributions $\mathrm { N } \left( \mu _ { x } , \sigma _ { x } ^ { 2 } \right)$ and $\mathrm { N } \left( \mu _ { y } , \sigma _ { y } ^ { 2 } \right)$ respectively, and that $W$ is a random variable such that $W = X + Y$.\\
(i) Use moment generating functions to show that the distribution of $W$ is $\mathrm { N } \left( \mu _ { x } + \mu _ { y } , \sigma _ { x } ^ { 2 } + \sigma _ { y } ^ { 2 } \right)$.\\
(ii) State the distribution of $X - Y$.

The diameters of the central poles of one brand of rotary clothes lines are normally distributed with mean 3.75 cm and variance $0.000125 \mathrm {~cm} ^ { 2 }$. The diameters of the cylindrical tubes, into which the central poles fit, are normally distributed with mean 3.85 cm and variance $0.0001 \mathrm {~cm} ^ { 2 }$. Poles and tubes are chosen at random. The 'clearance' between a tube and a pole is the diameter of the tube minus the diameter of the pole.\\
(iii) Find the probability that a pole and tube have a clearance between 0.08 cm and 0.13 cm .\\
(iv) Given that a pole and tube have a clearance between 0.08 cm and 0.13 cm , find the probability that the clearance is between 0.11 cm and 0.125 cm .

\hfill \mbox{\textit{Pre-U Pre-U 9795/2 2013 Q4}}
This paper (12 questions)
View full paper
Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 Q12