Probability density and cumulative distribution functionsEdexcel A-Level Further Maths: Subtopic test

1. 1
   The continuous random variable $X$ has probability density function $\mathrm{f}(x)=kx^2$ for $0\leq x\leq3$, and $\mathrm{f}(x)=0$ otherwise, where $k$ is a constant.

   (a)

   Find the value of $k$.

   [1 mark]

   • A$\frac19$
   • B$\frac13$
   • C$\frac1{27}$
   • D$9$

   (b)

   Find $\mathrm{P}(X>2)$.

   [1 mark]

   • A$\frac{8}{27}$
   • B$\frac13$
   • C$\frac59$
   • D$\frac{19}{27}$

   (c)

   Find the cumulative distribution function $\mathrm{F}(x)$ for $0\leq x\leq3$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The continuous random variable $X$ has cumulative distribution function $\mathrm{F}(x)=0$ for $x<1$, $\mathrm{F}(x)=\dfrac{x^2-1}{8}$ for $1\leq x\leq3$, and $\mathrm{F}(x)=1$ for $x>3$.

   (a)

   Find $\mathrm{P}(X\leq2)$.

   [1 mark]

   • A$\frac12$
   • B$\frac34$
   • C$\frac38$
   • D$\frac18$

   (b)

   Find the probability density function $\mathrm{f}(x)$ for $1\leq x\leq3$.

   [1 mark]

   • A$\dfrac{x^2-1}{8}$
   • B$\dfrac{x}{4}$
   • C$\dfrac{x^2}{4}$
   • D$\dfrac{x}{8}$

   (c)

   Find $\mathrm{P}(1.5<X\leq2.5)$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The time $X$, in hours, that a customer waits for a delivery has probability density function $\mathrm{f}(x)=kx$ for $0\leq x\leq2$, $\mathrm{f}(x)=2k$ for $2<x\leq4$, and $\mathrm{f}(x)=0$ otherwise, where $k$ is a constant.

   (a)

   Show that $k=\frac16$.

   [3 marks]

   (b)

   Find the probability that a customer waits between 1 and 3 hours.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The continuous random variable $X$ has probability density function $\mathrm{f}(x)=\dfrac{k}{\sqrt x}$ for $1\leq x\leq9$, and $\mathrm{f}(x)=0$ otherwise, where $k$ is a constant.

   (a)

   Find the value of $k$, and find the cumulative distribution function $\mathrm{F}(x)$ for all values of $x$.

   [6 marks]

   (b)

   Use $\mathrm{F}(x)=\dfrac{\sqrt x-1}{2}$ for $1\leq x\leq9$ to find $\mathrm{P}(X>4)$, and find the value $q$ such that $\mathrm{P}(X\leq q)=0.75$. Explain why $\mathrm{P}(X<4)=\mathrm{P}(X\leq4)$.

   [6 marks]

   Total for question 4: 12 marks

End of questions