Statistics 3 (S3): Combinations of random variablesEdexcel International A Level Further Maths: Topic test

1. 1
   The random variables $X$ and $Y$ are independent, with $X\sim\mathrm{N}(50,6^2)$ and $Y\sim\mathrm{N}(30,8^2)$.

   (a)

   What is the distribution of $X+Y$?

   [1 mark]

   • A$\mathrm{N}(80,14)$
   • B$\mathrm{N}(80,10)$
   • C$\mathrm{N}(20,100)$
   • D$\mathrm{N}(80,100)$

   (b)

   What is the distribution of $X-Y$?

   [1 mark]

   • A$\mathrm{N}(20,28)$
   • B$\mathrm{N}(20,100)$
   • C$\mathrm{N}(80,100)$
   • D$\mathrm{N}(20,2)$

   (c)

   Find $\mathrm{P}(X<Y)$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The mass, $S$ grams, of a bag of sugar is modelled by $S\sim\mathrm{N}(1010,8^2)$. The masses of different bags are independent.

   (a)

   Four bags are chosen at random. What is the distribution of $T=S_1+S_2+S_3+S_4$, the total mass of the four bags?

   [1 mark]

   • A$\mathrm{N}(4040,256)$
   • B$\mathrm{N}(4040,1024)$
   • C$\mathrm{N}(4040,64)$
   • D$\mathrm{N}(1010,256)$

   (b)

   One bag is chosen at random and its mass is multiplied by $4$. What is the distribution of $4S$?

   [1 mark]

   • A$\mathrm{N}(4040,256)$
   • B$\mathrm{N}(4040,64)$
   • C$\mathrm{N}(4040,1024)$
   • D$\mathrm{N}(1010,1024)$

   (c)

   Find the probability that the total mass of four randomly chosen bags exceeds $4050$ g.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A product is made in two stages that are carried out independently. The time, $A$ minutes, for stage one is modelled by $A\sim\mathrm{N}(12.5,1.8^2)$ and the time, $B$ minutes, for stage two is modelled by $B\sim\mathrm{N}(8.4,1.2^2)$.

   (a)

   Find the probability that the two stages together take more than $24$ minutes.

   [3 marks]

   (b)

   Find the probability that stage one takes more than $5$ minutes longer than stage two.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The mass, $A$ grams, of an apple is modelled by $A\sim\mathrm{N}(150,12^2)$, and the masses of different apples are independent. An empty box has mass $B$ grams, where $B\sim\mathrm{N}(80,5^2)$, independent of the apples.

   (a)

   Six apples chosen at random are put into an empty box. Find the probability that the total mass of the box and the six apples exceeds $1000$ g.

   [6 marks]

   (b)

   Let $Y=A_1+A_2+A_3$ be the total mass of three different apples chosen at random, and let $Z=3A$ be three times the mass of a single apple chosen at random. Find $\mathrm{P}(Y<440)$ and $\mathrm{P}(Z<440)$, and explain why the two probabilities are different even though $Y$ and $Z$ have the same mean.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   The random variables $X$ and $Y$ are independent, with $X\sim\mathrm{N}(8,2^2)$ and $Y\sim\mathrm{N}(5,3^2)$.

   (a)

   What is the distribution of $3X+2$?

   [1 mark]

   • A$\mathrm{N}(24,36)$
   • B$\mathrm{N}(26,36)$
   • C$\mathrm{N}(26,14)$
   • D$\mathrm{N}(26,6)$

   (b)

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

   [1 mark]

   • A$\mathrm{N}(-2,22)$
   • B$\mathrm{N}(-2,32)$
   • C$\mathrm{N}(-2,13)$
   • D$\mathrm{N}(-2,40)$

   (c)

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

   [2 marks]

   Total for question 5: 4 marks

6. 6
   The times, in seconds, taken by two runners over $100$ m are modelled by independent random variables $A\sim\mathrm{N}(12.4,0.3^2)$ for the first runner and $B\sim\mathrm{N}(12.1,0.4^2)$ for the second.

   (a)

   What is the distribution of $A-B$?

   [1 mark]

   • A$\mathrm{N}(0.3,0.07)$
   • B$\mathrm{N}(0.3,0.5)$
   • C$\mathrm{N}(0.3,0.25)$
   • D$\mathrm{N}(24.5,0.25)$

   (b)

   Find the probability that the first runner is faster than the second, that is $\mathrm{P}(A<B)$.

   [1 mark]

   • A$0.274$
   • B$0.726$
   • C$0.115$
   • D$0.382$

   (c)

   Find the probability that the first runner takes more than $0.8$ seconds longer than the second runner.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A ferry carries vehicles whose masses are independent. The mass of a car, in kg, is modelled by $C\sim\mathrm{N}(1350,150^2)$ and the mass of a van, in kg, is modelled by $V\sim\mathrm{N}(2100,200^2)$.

   (a)

   Three cars are chosen at random. Find the probability that their total mass exceeds $4200$ kg.

   [3 marks]

   (b)

   Three cars and one van are chosen at random. Find the probability that their total mass exceeds $6500$ kg.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A jug is filled with a volume $J$ ml of cordial, where $J\sim\mathrm{N}(1000,10^2)$. Cups are filled with volumes $C$ ml, where $C\sim\mathrm{N}(240,6^2)$. All the volumes are independent.

   (a)

   The cordial from one jug is poured into four randomly chosen cups. Find the probability that there is enough cordial in the jug to fill all four cups, that is $\mathrm{P}(J>C_1+C_2+C_3+C_4)$.

   [6 marks]

   (b)

   Find the volume $v$ ml, to the nearest ml, such that the total volume of four randomly chosen cups exceeds $v$ with probability $0.01$.

   [6 marks]

   Total for question 8: 12 marks

End of questions