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

1. 1
   The mass of an apple, $A$ g, is modelled by $A\sim N(150, 12^2)$ and the mass of a pear, $P$ g, by $P\sim N(200, 15^2)$. The masses of apples and pears are independent.

   (a)

   The total mass $T=A+P$ of one apple and one pear has distribution

   [1 mark]

   • A$N(350, 27)$
   • B$N(350, 27^2)$
   • C$N(350, 81)$
   • D$N(350, 369)$

   (b)

   The difference in mass $D=P-A$ has distribution

   [1 mark]

   • A$N(50, 81)$
   • B$N(350, 369)$
   • C$N(50, 369)$
   • D$N(-50, 369)$

   (c)

   Find the probability that a randomly chosen pear is heavier than a randomly chosen apple.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The independent random variables $X$ and $Y$ are distributed as $X\sim N(20, 3^2)$ and $Y\sim N(15, 2^2)$. The random variable $W=3X-2Y$.

   (a)

   Find $\mathrm{E}(W)$.

   [1 mark]

   • A$90$
   • B$30$
   • C$5$
   • D$60$

   (b)

   Find $\mathrm{Var}(W)$.

   [1 mark]

   • A$97$
   • B$65$
   • C$13$
   • D$35$

   (c)

   Find $\mathrm{P}(W<25)$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The length of a metal rod, $X$ cm, is modelled by $X\sim N(50, 0.4^2)$. The lengths of different rods are independent.

   (a)

   Two rods are chosen at random and placed end to end. Find the probability that their total length exceeds $100.5$ cm.

   [3 marks]

   (b)

   A manufacturer makes a long piece by doubling the length of one randomly chosen rod, so that the length of the piece is $2X$. Find $\mathrm{P}(2X>100.5)$ and explain why it differs from your answer to (a).

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The mass of an adult man, $M$ kg, is modelled by $M\sim N(80, 8^2)$ and the mass of an adult woman, $W$ kg, by $W\sim N(65, 6^2)$. The masses of different people are independent.

   (a)

   A lift carries three men and two women, chosen at random. Find the probability that their total mass exceeds $400$ kg.

   [6 marks]

   (b)

   (i) Find the probability that a randomly chosen man is heavier than a randomly chosen woman.
   (ii) Find the probability that the mean mass of four randomly chosen men exceeds the mass of a randomly chosen woman by more than $10$ kg.

   [6 marks]

   Total for question 4: 12 marks

End of questions