Discrete uniform distributionAQA A-Level Further Maths: Subtopic test

1. 1
   A fair eight-sided die, numbered $1$ to $8$, is rolled once. The score $X$ is the number on the face it lands on, and $X$ is modelled by the discrete uniform distribution on $\{1,2,\ldots,8\}$.

   (a)

   Find $P(X>5)$.

   [1 mark]

   • A$\frac38$
   • B$\frac14$
   • C$\frac12$
   • D$\frac58$

   (b)

   Find $E(X)$.

   [1 mark]

   • A$4$
   • B$4.5$
   • C$3.5$
   • D$36$

   (c)

   Find the variance of $X$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The discrete random variable $X$ has a discrete uniform distribution on $\{1,2,\ldots,n\}$, where $n$ is a positive integer. It is given that $E(X)=10.5$.

   (a)

   Find the value of $n$.

   [1 mark]

   • A$21$
   • B$10.5$
   • C$20$
   • D$22$

   (b)

   Find the variance of $X$.

   [1 mark]

   • A$\frac{100}{3}$
   • B$5.77$
   • C$399$
   • D$\frac{133}{4}$

   (c)

   Find $P(X>E(X))$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The discrete random variable $X$ has a discrete uniform distribution on $\{1,2,\ldots,n\}$, so that $P(X=x)=\frac1n$ for $x=1,2,\ldots,n$.

   (a)

   Prove that $E(X)=\frac{n+1}{2}$.

   [3 marks]

   (b)

   Given that $\sum_{x=1}^{n}x^2=\frac{n(n+1)(2n+1)}{6}$, prove that $\mathrm{Var}(X)=\frac{n^2-1}{12}$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A school raffle uses tickets numbered $1$ to $n$. One ticket is drawn at random and $X$ is its number, modelled by the discrete uniform distribution on $\{1,2,\ldots,n\}$.

   (a)

   This year $n=30$. (i) State two features of the situation that justify using the discrete uniform model. (ii) Find the probability that the number drawn is a multiple of $4$. (iii) Find $E(X)$ and $\mathrm{Var}(X)$.

   [6 marks]

   (b)

   Next year $n$ will be increased so that the standard deviation of $X$ is greater than $10$. (i) Find the smallest possible value of $n$. (ii) For this value of $n$, find $E(X)$ and $P(X<E(X))$.

   [6 marks]

   Total for question 4: 12 marks

End of questions