Further Statistics 1: Discrete probability distributionsEdexcel A-Level Further Maths: Topic test

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

   (a)

   What is the value of $k$?

   [1 mark]

   • A$0.1$
   • B$0.25$
   • C$0.05$
   • D$0.4$

   (b)

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

   [1 mark]

   • A$2.5$
   • B$1$
   • C$2$
   • D$3$

   (c)

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

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The discrete random variable $W$ takes the values $-2$, $0$, $1$ and $3$ with probabilities $0.15$, $0.35$, $0.3$ and $0.2$ respectively.

   (a)

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

   [1 mark]

   • A$0.5$
   • B$1.2$
   • C$0.9$
   • D$0.6$

   (b)

   What is $\mathrm{E}(W^2)$?

   [1 mark]

   • A$0.36$
   • B$2.7$
   • C$0.6$
   • D$3.3$

   (c)

   A game pays the player $\pounds(3W^2-2)$. Find the expected payment.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The discrete random variable $X$ is the number of goals scored by a youth football team in a match. $P(X=0)=0.1$, $P(X=1)=p$, $P(X=2)=q$ and $P(X=3)=0.3$, and $\mathrm{E}(X)=1.7$.

   (a)

   Find the values of $p$ and $q$.

   [3 marks]

   (b)

   (i) Find $\mathrm{Var}(X)$.
   (ii) Over a season of 38 matches, the team's goals per match have mean $1.68$ and variance $1.04$. Comment on the suitability of the model.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The discrete random variable $Y$ has probability function $P(Y=y)=\dfrac{y^2}{c}$ for $y=1,2,3,4$, where $c$ is a constant.

   (a)

   Find the value of $c$, the value of $\mathrm{E}(Y)$ and the value of $\mathrm{Var}(Y)$.

   [6 marks]

   (b)

   $Y$ is the number of hundreds of loaves a bakery sells in a day. The profit, in pounds, on a day is $60Y-12Y^2$.
   (i) Find the expected daily profit.
   (ii) Find the probability that the profit on a day is at least $\pounds72$.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   A hire company records the number of days, $D$, that a customer returns a van late. $P(D=0)=0.5$, $P(D=1)=0.25$, $P(D=2)=0.15$ and $P(D=3)=0.1$.

   (a)

   What is the probability that a van is returned at least $2$ days late?

   [1 mark]

   • A$0.15$
   • B$0.5$
   • C$0.25$
   • D$0.75$

   (b)

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

   [1 mark]

   • A$0.85$
   • B$1.5$
   • C$0.55$
   • D$1.75$

   (c)

   The company charges $\pounds(20+15D)$ for a hire. Find the expected charge.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   The discrete random variable $T$ has probability function $P(T=t)=\dfrac{t+1}{20}$ for $t=1,2,3,4,5$.

   (a)

   What is $P(T\ge4)$?

   [1 mark]

   • A$\frac{6}{20}$
   • B$\frac{11}{20}$
   • C$\frac{9}{20}$
   • D$\frac{15}{20}$

   (b)

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

   [1 mark]

   • A$3$
   • B$2.5$
   • C$14$
   • D$3.5$

   (c)

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

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A teacher records the number of correct answers, $X$, in a three-question quiz for a large class, so $X$ takes the values $0,1,2,3$. She considers two models. In Model A the four values are equally likely. In Model B, $P(X=0)=0.1$, $P(X=1)=0.2$, $P(X=2)=0.4$ and $P(X=3)=0.3$.

   (a)

   Find the mean and variance of $X$ under Model A.

   [3 marks]

   (b)

   The marks for the whole class have mean $1.88$ and variance $0.91$. Find the mean and variance of $X$ under Model B, and state, with a reason, which model is the more suitable.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A discrete random variable $X$ takes the values $1$, $2$ and $3$ with $P(X=1)=a$, $P(X=2)=b$ and $P(X=3)=c$. It is given that $\mathrm{E}(X)=2.1$ and $\mathrm{Var}(X)=0.49$.

   (a)

   Find the values of $a$, $b$ and $c$.

   [6 marks]

   (b)

   In a game a player pays $\pounds4.50$ to play and then receives $\pounds X^2$.
   (i) Find the expected profit of the player per game.
   (ii) Find the variance of the amount the player receives.

   [6 marks]

   Total for question 8: 12 marks

End of questions