S1: Discrete random variablesEdexcel International A Level Maths: Topic test

1. 1
   The discrete random variable $X$ has probability function $\mathrm{P}(X=x)=c\times2^{x}$ for $x=0,1,2,3$, where $c$ is a constant.

   (a)

   Find the value of $c$.

   [1 mark]

   • A$\frac18$
   • B$\frac1{15}$
   • C$\frac1{14}$
   • D$\frac16$

   (b)

   Find $\mathrm{P}(X\le1)$.

   [1 mark]

   • A$\frac1{15}$
   • B$\frac2{15}$
   • C$\frac15$
   • D$\frac7{15}$

   (c)

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

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A fair spinner has twelve equal sectors numbered 1 to 12. The random variable $S$ is the number on the sector on which it lands.

   (a)

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

   [1 mark]

   • A$6$
   • B$7$
   • C$78$
   • D$6.5$

   (b)

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

   [1 mark]

   • A$\frac{143}{12}$
   • B$12$
   • C$\frac{169}{12}$
   • D$\frac{13}{2}$

   (c)

   A player is awarded $(2S-5)$ points. Find the expected number of points awarded.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The discrete random variable $Y$ takes the values 2, 4, 6, 8 and 10. Its cumulative distribution function satisfies $\mathrm{F}(2)=0.1$, $\mathrm{F}(4)=0.3$, $\mathrm{F}(6)=0.6$, $\mathrm{F}(8)=0.9$ and $\mathrm{F}(10)=1$.

   (a)

   Find $\mathrm{P}(Y=8)$ and $\mathrm{P}(4<Y\le8)$.

   [3 marks]

   (b)

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

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A prize draw uses 20 balls numbered 1 to 20, and one ball is chosen at random. The random variable $R$ is the number on the ball chosen. The prize is $C$ pounds, where $C=4R-6$.

   (a)

   (i) Find $\mathrm{E}(R)$ and $\mathrm{Var}(R)$.
   (ii) Find $\mathrm{E}(C)$ and $\mathrm{Var}(C)$.

   [6 marks]

   (b)

   (i) Find $\mathrm{P}(C>50)$.
   (ii) The organiser charges an entry fee of $f$ pounds per draw, so that the expected profit per draw is 2 pounds. Find $f$.
   (iii) Find the standard deviation of the organiser's profit per draw, giving your answer to 3 significant figures.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   The discrete random variable $X$ has $\mathrm{E}(X)=-2$ and $\mathrm{E}(X^2)=13$.

   (a)

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

   [1 mark]

   • A$9$
   • B$11$
   • C$17$
   • D$15$

   (b)

   Find $\mathrm{Var}(5-2X)$.

   [1 mark]

   • A$9$
   • B$18$
   • C$41$
   • D$36$

   (c)

   Find $\mathrm{E}\big((X+3)^2\big)$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   The discrete random variable $X$ takes the values 1, 2, 3, 4 and 5, and its cumulative distribution function is $\mathrm{F}(x)=\frac{x^2}{25}$ for these values of $x$.

   (a)

   Find $\mathrm{P}(X=3)$.

   [1 mark]

   • A$\frac{9}{25}$
   • B$\frac15$
   • C$\frac{3}{25}$
   • D$\frac{13}{25}$

   (b)

   Find $\mathrm{P}(X\ge4)$.

   [1 mark]

   • A$\frac{7}{25}$
   • B$\frac{9}{25}$
   • C$\frac{16}{25}$
   • D$\frac{21}{25}$

   (c)

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

   [2 marks]

   Total for question 6: 4 marks

7. 7
   Raffle tickets are numbered from 1 to $n$, and one ticket is drawn at random. The random variable $X$ is the number on the ticket drawn. It is given that $\mathrm{P}(X\le5)=\frac16$.

   (a)

   Find the value of $n$ and hence find $\mathrm{P}(X>24)$.

   [3 marks]

   (b)

   Using your value of $n$, find $\mathrm{E}(X)$ and $\mathrm{Var}(X)$. Hence find $\mathrm{Var}(4-3X)$.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   The discrete random variable $X$ has $\mathrm{P}(X=0)=a$, $\mathrm{P}(X=1)=0.3$, $\mathrm{P}(X=2)=b$ and $\mathrm{P}(X=3)=0.2$, where $a$ and $b$ are constants. It is given that $\mathrm{E}(X)=1.5$.

   (a)

   (i) Find the values of $a$ and $b$.
   (ii) Write down $\mathrm{F}(2)$ and find $\mathrm{P}(1\le X<3)$.

   [6 marks]

   (b)

   (i) Show that $\mathrm{Var}(X)=1.05$.
   (ii) The random variable $Y=6-4X$. Find $\mathrm{E}(Y)$ and $\mathrm{Var}(Y)$.

   [6 marks]

   Total for question 8: 12 marks

End of questions