Further Statistics 1: Discrete random variablesAQA A-Level Further Maths: Topic test

1. 1
   The discrete random variable $X$ takes the values $0,1,2,3$ with $P(X=0)=0.15$, $P(X=1)=p$, $P(X=2)=2p$ and $P(X=3)=0.25$.

   (a)

   What is the value of $p$?

   [1 mark]

   • A$0.3$
   • B$0.6$
   • C$0.2$
   • D$0.4$

   (b)

   What is $\mathrm{E}(X)$?

   [1 mark]

   • A$1.75$
   • B$1.5$
   • C$1.0$
   • D$4.05$

   (c)

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

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A taxi driver's income $F$ pounds from one journey is modelled by $F=2.5D+4$, where $D$ is the length of the journey in whole miles. $D$ is a discrete random variable with $\mathrm{E}(D)=6.4$ and $\mathrm{Var}(D)=3.6$.

   (a)

   What is $\mathrm{E}(F)$?

   [1 mark]

   • A$16$
   • B$20$
   • C$10.4$
   • D$22.5$

   (b)

   What is $\mathrm{Var}(F)$?

   [1 mark]

   • A$9.0$
   • B$26.5$
   • C$7.6$
   • D$22.5$

   (c)

   The cost to the driver of a journey is £$(0.9D+1.5)$. Find the standard deviation of the driver's profit on a journey, which is the income minus the cost.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The number of people $X$ in a lottery syndicate takes the values $1,2,4$ and $5$ with probabilities $0.2$, $0.3$, $0.4$ and $0.1$ respectively. A prize of £10 is shared equally among the members, so each member receives £$S$, where $S=\dfrac{10}{X}$.

   (a)

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

   [3 marks]

   (b)

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

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A random number generator is designed to select a number $X$ from $\{1,2,\ldots,n\}$ so that each number is equally likely, where $n$ is a positive integer. It is known that $\mathrm{Var}(X)=8.25$.

   (a)

   Find the value of $n$, the value of $\mathrm{E}(X)$ and the probability that $X$ is greater than $7$.

   [6 marks]

   (b)

   A game pays a player £$W$, where $W=2X-7$. Find $\mathrm{E}(W)$ and $\mathrm{Var}(W)$. A second game pays £$X^{2}$. Find the expected payment from the second game.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   The discrete random variable $X$ has probability function $P(X=x)=k(6-x)$ for $x=1,2,3,4,5$, and $P(X=x)=0$ otherwise, where $k$ is a constant.

   (a)

   What is the value of $k$?

   [1 mark]

   • A$\dfrac{1}{21}$
   • B$\dfrac{1}{5}$
   • C$\dfrac{1}{25}$
   • D$\dfrac{1}{15}$

   (b)

   What is $P(X\geq3)$?

   [1 mark]

   • A$0.6$
   • B$0.2$
   • C$0.4$
   • D$0.667$

   (c)

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

   [2 marks]

   Total for question 5: 4 marks

6. 6
   An online seller receives a weekly number of orders $X$ with $\mathrm{E}(X)=40$ and $\mathrm{Var}(X)=36$. The weekly profit in pounds is $Y=15X-120$.

   (a)

   What is $\mathrm{E}(Y)$?

   [1 mark]

   • A$480$
   • B$600$
   • C$720$
   • D$-80$

   (b)

   What is the standard deviation of $Y$?

   [1 mark]

   • A$8100$
   • B$90$
   • C$6$
   • D$225$

   (c)

   Find $\mathrm{E}(X^{2})$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   The discrete random variable $X$ takes the values $1$, $3$ and $5$ with $P(X=1)=0.5$, $P(X=3)=0.3$ and $P(X=5)=0.2$.

   (a)

   Find $\mathrm{E}(5X^{3})$.

   [3 marks]

   (b)

   Find $\mathrm{Var}(X^{2})$.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A fair five-sided spinner is numbered $1,2,3,4,5$. The score $X$ is the number on which it lands, so $X$ has a discrete uniform distribution on $\{1,2,3,4,5\}$.

   (a)

   Show, using the definitions of expectation and variance, that $\mathrm{E}(X)=3$ and $\mathrm{Var}(X)=2$. Hence find the probability that the score is within one standard deviation of the mean.

   [6 marks]

   (b)

   A player pays £10 to play and receives £$3X$, so the profit is $V=3X-10$ pounds. Find $\mathrm{E}(V)$, the standard deviation of $V$, and the probability that the player makes a profit.

   [6 marks]

   Total for question 8: 12 marks

End of questions