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

1. 1
   The continuous random variable $X$ is uniformly distributed over the interval $[2,8]$.

   (a)

   Find the value of the probability density function $f(x)$ for $2\le x\le8$.

   [1 mark]

   • A$\frac16$
   • B$\frac18$
   • C$\frac12$
   • D$6$

   (b)

   Find $P(X\le3.5)$.

   [1 mark]

   • A$\frac{7}{16}$
   • B$\frac16$
   • C$\frac34$
   • D$\frac14$

   (c)

   Find $E(X)$ and $\text{Var}(X)$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The length $L$ cm of a rod cut by a machine is modelled by a continuous uniform distribution over the interval $[49,51]$.

   (a)

   Find the probability that a rod is longer than 50.3 cm.

   [1 mark]

   • A$0.65$
   • B$0.35$
   • C$0.3$
   • D$0.7$

   (b)

   Find $\text{Var}(L)$.

   [1 mark]

   • A$\frac16$
   • B$\frac1{12}$
   • C$\frac13$
   • D$2$

   (c)

   The machine's target length is 50 cm. Find the probability that a rod is within 0.25 cm of the target.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The continuous random variable $X$ is uniformly distributed over the interval $[a,b]$, where $b>a$.

   (a)

   Show that the cumulative distribution function is $F(x)=\frac{x-a}{b-a}$ for $a\le x\le b$.

   [3 marks]

   (b)

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

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The continuous random variable $X$ is uniformly distributed over the interval $[a,b]$, where $b>a$. You may use the result $E(X)=\frac{a+b}{2}$.

   (a)

   Show that $\text{Var}(X)=\frac{(b-a)^2}{12}$.

   [6 marks]

   (b)

   Given that $E(X)=12$ and $\text{Var}(X)=12$:
   (i) find the values of $a$ and $b$;
   (ii) find $P(X>15)$.

   [6 marks]

   Total for question 4: 12 marks

End of questions