Functions of a DRVAQA A-Level Further Maths: Subtopic test

1. 1
   The discrete random variable $X$ takes the values $1,2,3$ with $P(X=1)=0.2$, $P(X=2)=0.5$ and $P(X=3)=0.3$.

   (a)

   Find $E(X^2)$.

   [1 mark]

   • A$4.41$
   • B$4.9$
   • C$\frac{14}{3}$
   • D$2.1$

   (b)

   Find $E\!\left(\frac{10}{X}\right)$.

   [1 mark]

   • A$5.5$
   • B$4.76$
   • C$0.55$
   • D$6.11$

   (c)

   Find $E(5X^3)$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The discrete random variable $X$ takes the values $1,2,4$ with $P(X=1)=0.4$, $P(X=2)=0.4$ and $P(X=4)=0.2$.

   (a)

   Find $E(6X^{-1})$.

   [1 mark]

   • A$3.0$
   • B$12$
   • C$0.65$
   • D$3.9$

   (b)

   Find $\mathrm{Var}(X^{-1})$.

   [1 mark]

   • A$0.5125$
   • B$0.4225$
   • C$0.09$
   • D$0.3$

   (c)

   Hence find $\mathrm{Var}(6X^{-1})$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The discrete random variable $X$ has probability distribution $P(X=x)=\frac{x}{10}$ for $x=1,2,3,4$.

   (a)

   Find $E(18X^{-3})$.

   [3 marks]

   (b)

   Given that $E(X^3)=35.4$, find $\mathrm{Var}(5X^3)$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A tile-maker cuts square tiles whose side length $X$ cm is a discrete random variable with $P(X=2)=0.25$, $P(X=3)=0.5$ and $P(X=4)=0.25$. The area of a tile is $A=X^2$ cm$^2$.

   (a)

   (i) Find $E(X)$. (ii) Find $E(A)$ and explain why it differs from $[E(X)]^2$. (iii) Find $\mathrm{Var}(A)$.

   [6 marks]

   (b)

   The cost, in pounds, of making a tile is $C=0.4A+1$. Find $E(C)$ and the standard deviation of $C$, and find the probability that the cost of a tile exceeds $E(C)$.

   [6 marks]

   Total for question 4: 12 marks

End of questions