Linear functions of a DRVAQA A-Level Further Maths: Subtopic test

1. 1
   The discrete random variable $X$ has $E(X)=5$ and $\mathrm{Var}(X)=4$. The random variable $Y$ is defined by $Y=3X+2$.

   (a)

   Find $E(Y)$.

   [1 mark]

   • A$17$
   • B$15$
   • C$21$
   • D$7$

   (b)

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

   [1 mark]

   • A$12$
   • B$38$
   • C$6$
   • D$36$

   (c)

   Find the standard deviation of $2-4X$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The number of special meals, $X$, sold at a café in one day has $E(X)=8.5$ and $\mathrm{Var}(X)=2.25$. The daily profit, in pounds, is $T=6X-20$.

   (a)

   Find the expected daily profit $E(T)$.

   [1 mark]

   • A$\pounds51$
   • B$-\pounds69$
   • C$\pounds31$
   • D$-\pounds11.5$

   (b)

   Find the standard deviation of the daily profit.

   [1 mark]

   • A$\pounds81$
   • B$\pounds9$
   • C$\pounds13.5$
   • D$\pounds1.5$

   (c)

   The café changes its pricing so that the daily profit becomes $T=8X-30$. Find $E(T)$ and $\mathrm{Var}(T)$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The noon temperature $C$ ($^\circ$C) in a greenhouse is modelled as a discrete random variable with $E(C)=22$ and standard deviation $3$. The temperature in degrees Fahrenheit is $F=1.8C+32$.

   (a)

   Find $E(F)$ and $\mathrm{Var}(F)$.

   [3 marks]

   (b)

   The reading $Y=aC+b$, with $a>0$, is standardised so that $E(Y)=0$ and $\mathrm{Var}(Y)=1$. Find the values of $a$ and $b$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The discrete random variable $X$ has distribution $P(X=1)=0.3$, $P(X=2)=0.4$, $P(X=3)=0.2$ and $P(X=4)=0.1$. The random variable $Y$ is defined by $Y=5-2X$.

   (a)

   (i) Find $E(X)$ and $\mathrm{Var}(X)$. (ii) Hence find $E(Y)$ and $\mathrm{Var}(Y)$.

   [6 marks]

   (b)

   Find the probability distribution of $Y$ and use it to calculate $E(Y)$ and $\mathrm{Var}(Y)$ directly. State whether your results agree with part (a).

   [6 marks]

   Total for question 4: 12 marks

End of questions