Mean and variance of discrete distributionsEdexcel A-Level Further Maths: Subtopic test

1. 1
   The discrete random variable $X$ has $P(X=1)=0.1$, $P(X=2)=0.3$, $P(X=3)=0.4$ and $P(X=4)=0.2$.

   (a)

   Find $E(X)$.

   [1 mark]

   • A$2.5$
   • B$0.675$
   • C$8.1$
   • D$2.7$

   (b)

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

   [1 mark]

   • A$8.1$
   • B$0.81$
   • C$0.9$
   • D$5.4$

   (c)

   Find $E\left(\frac1X\right)$.

   [2 marks]

   Total for question 1: 4 marks

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

   (a)

   Find the value of $k$.

   [1 mark]

   • A$\frac1{15}$
   • B$\frac15$
   • C$\frac1{10}$
   • D$\frac1{55}$

   (b)

   Find $E(Y)$.

   [1 mark]

   • A$3$
   • B$15$
   • C$\frac{11}{3}$
   • D$1$

   (c)

   Find $\text{Var}(Y)$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A fairground game costs £3 to play. A player is paid £20 with probability $0.05$, £6 with probability $0.2$ and nothing otherwise. Let $X$ be the player's profit in pounds, where profit is the payout minus the £3 cost.

   (a)

   Find $E(X)$ and interpret your answer in context.

   [3 marks]

   (b)

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

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A manager models the number of delivery vans, $X$, arriving at a depot in a 10-minute period by $P(X=0)=0.1$, $P(X=1)=0.4$, $P(X=2)=0.3$ and $P(X=3)=0.2$. Records over many periods give a sample mean of $1.9$ vans and a sample standard deviation of $1.3$ vans. The cost in pounds of handling the vans in a period is $C=5X^2+10$. A cheaper contract charges a flat £25 per period.

   (a)

   (i) Find $E(X)$ and $\text{Var}(X)$ for the manager's model.
   (ii) Compare the model with the records and comment on its suitability.

   [6 marks]

   (b)

   Use the manager's model to find $E(C)$, and decide which contract is cheaper on average. Comment on how the records affect your decision.

   [6 marks]

   Total for question 4: 12 marks

End of questions