Linear combinations of Normal random variablesEdexcel International A Level Further Maths: Subtopic test

1. 1
   The time $X$ minutes that a student takes to complete the first section of a test is modelled by $\mathrm{N}(20,4^2)$. The time $Y$ minutes taken to complete the second section is modelled by $\mathrm{N}(15,3^2)$. $X$ and $Y$ are independent.

   (a)

   What is the distribution of the total time $X+Y$?

   [1 mark]

   • A$\mathrm{N}(35,5)$
   • B$\mathrm{N}(35,25)$
   • C$\mathrm{N}(35,49)$
   • D$\mathrm{N}(35,7)$

   (b)

   What is the distribution of the difference $X-Y$?

   [1 mark]

   • A$\mathrm{N}(5,25)$
   • B$\mathrm{N}(5,7)$
   • C$\mathrm{N}(5,1)$
   • D$\mathrm{N}(35,25)$

   (c)

   Find the probability that the first section takes longer than the second.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The random variables $X$ and $Y$ are independent, with $X\sim\mathrm{N}(12,3^2)$ and $Y\sim\mathrm{N}(7,2^2)$.

   (a)

   What is the distribution of $3X-2Y$?

   [1 mark]

   • A$\mathrm{N}(22,65)$
   • B$\mathrm{N}(22,25)$
   • C$\mathrm{N}(50,97)$
   • D$\mathrm{N}(22,97)$

   (b)

   $X_1$ and $X_2$ are independent observations of $X$. What is the distribution of $X_1-X_2$?

   [1 mark]

   • A$\mathrm{N}(0,0)$
   • B$\mathrm{N}(0,9)$
   • C$\mathrm{N}(0,18)$
   • D$\mathrm{N}(0,36)$

   (c)

   Find $\mathrm{P}(3X-2Y>30)$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Four adult passengers, chosen at random, ride in a lift. The mass of an adult passenger is modelled by $\mathrm{N}(75,12^2)$ kg and the mass of the luggage that one passenger carries by $\mathrm{N}(18,5^2)$ kg. All the masses are independent. A calculator may be used.

   (a)

   Find the probability that the total mass of the four passengers, without their luggage, exceeds $320$ kg.

   [3 marks]

   (b)

   Each passenger also carries one piece of luggage. Find the probability that the total mass of the four passengers and their luggage exceeds $400$ kg.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   An engineer fits metal rods into slots. The length $R$ cm of a rod is modelled by $\mathrm{N}(20,0.12^2)$ and the width $S$ cm of a slot by $\mathrm{N}(20.3,0.1^2)$. All lengths and widths are independent. A calculator may be used.

   (a)

   A rod and a slot are chosen at random. Let $C=S-R$ be the clearance.
   (i) Find the distribution of $C$.
   (ii) Find the probability that the rod is too long for the slot, that is $C<0$.

   [6 marks]

   (b)

   Five rods are chosen at random, with lengths $R_1,\dots,R_5$, and laid end to end to form a bar of length $T=R_1+R_2+R_3+R_4+R_5$. A second bar is made by choosing one rod at random and using five times its length, $5R$.
   (i) Find $\mathrm{P}(T>100.5)$.
   (ii) Find $\mathrm{P}(5R>100.5)$.
   (iii) Explain why the two probabilities are different.

   [6 marks]

   Total for question 4: 12 marks

End of questions