The continuous uniform distributionEdexcel A-Level Further Maths: Subtopic test

1. 1
   The continuous random variable $X$ is uniformly distributed over the interval $[2,10]$.

   (a)

   Find $\mathrm{P}(X>7)$.

   [1 mark]

   • A$\frac{3}{10}$
   • B$\frac{5}{8}$
   • C$\frac{3}{8}$
   • D$\frac{1}{8}$

   (b)

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

   [1 mark]

   • A$\frac{16}{3}$
   • B$\frac{2}{3}$
   • C$6$
   • D$12$

   (c)

   Given that $\mathrm{P}(X<c)=0.35$, find the value of $c$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The continuous random variable $Y$ is uniformly distributed over the interval $[a,b]$. The mean of $Y$ is $9$ and the variance of $Y$ is $12$.

   (a)

   Find the value of $b-a$.

   [1 mark]

   • A$6$
   • B$144$
   • C$\sqrt{12}$
   • D$12$

   (b)

   Find the value of $a$.

   [1 mark]

   • A$15$
   • B$3$
   • C$-3$
   • D$9$

   (c)

   Find $\mathrm{P}(Y>11)$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The continuous random variable $X$ has a continuous uniform distribution over the interval $[a,b]$, where $a<b$.

   (a)

   Show by integration that $\mathrm{E}(X)=\frac{a+b}{2}$.

   [3 marks]

   (b)

   Derive the cumulative distribution function $F(x)$ of $X$, for all real $x$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A coffee machine dispenses a volume $V$ ml of coffee into each cup. The volume is modelled as being uniformly distributed over the interval $[190,210]$.

   (a)

   (i) Find $\mathrm{E}(V)$ and $\mathrm{Var}(V)$.
   (ii) Find $\mathrm{P}(V<193)$.
   (iii) A cup is underfilled if it contains less than 193 ml. Find the probability that, in a random sample of 5 cups, exactly one is underfilled.

   [6 marks]

   (b)

   The machine is adjusted so that $V$ is now uniformly distributed over $[a,b]$, with mean $200$ and $\mathrm{P}(V<193)=0.1$. Find the values of $a$ and $b$, and hence find the variance of $V$.

   [6 marks]

   Total for question 4: 12 marks

End of questions