Statistics 2 (S2): Continuous random variablesEdexcel International A Level Further Maths: Topic test

1. 1
   The continuous random variable $X$ has probability density function $f(x)=k(x+1)$ for $0\le x\le2$, and $f(x)=0$ otherwise, where $k$ is a constant.

   (a)

   Find the value of $k$.

   [1 mark]

   • A$\frac13$
   • B$\frac16$
   • C$\frac14$
   • D$\frac12$

   (b)

   Find $\mathrm{P}(X>1)$.

   [1 mark]

   • A$\frac58$
   • B$\frac38$
   • C$\frac12$
   • D$\frac34$

   (c)

   Find the cumulative distribution function $\mathrm{F}(x)$ for $0\le x\le2$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The thickness, $Y$ mm, of a protective coating has cumulative distribution function $\mathrm{F}(y)=0$ for $y<0$, $\mathrm{F}(y)=\frac{3y^2-y^3}{4}$ for $0\le y\le2$, and $\mathrm{F}(y)=1$ for $y>2$.

   (a)

   Find $\mathrm{P}(Y>1.5)$.

   [1 mark]

   • A$\frac{27}{32}$
   • B$\frac{5}{32}$
   • C$\frac{9}{16}$
   • D$\frac14$

   (b)

   Which expression is the probability density function $f(y)$ for $0\le y\le2$?

   [1 mark]

   • A$\frac{3y^2-y^3}{4}$
   • B$\frac{6y-y^2}{4}$
   • C$\frac{y^3}{4}-\frac{y^4}{16}$
   • D$\frac{3y(2-y)}{4}$

   (c)

   Show that the mode of $Y$ is $1$ mm.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The continuous random variable $X$ has probability density function $f(x)=ax$ for $0\le x<1$, $f(x)=a$ for $1\le x\le3$, and $f(x)=0$ otherwise, where $a$ is a constant.

   (a)

   Show that $a=\frac25$.

   [3 marks]

   (b)

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

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The continuous random variable $X$ has probability density function $f(x)=k(x-1)(5-x)$ for $1\le x\le5$, and $f(x)=0$ otherwise, where $k$ is a constant.

   (a)

   Show that $k=\frac{3}{32}$ and hence find the cumulative distribution function $\mathrm{F}(x)$ for $1\le x\le5$.

   [6 marks]

   (b)

   Find the mode and the mean of $X$, and show that $\mathrm{Var}(X)=\frac45$.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   The continuous random variable $X$ has probability density function $f(x)=k(4-x^2)$ for $0\le x\le2$, and $f(x)=0$ otherwise, where $k$ is a constant.

   (a)

   Find the value of $k$.

   [1 mark]

   • A$\frac{3}{16}$
   • B$\frac18$
   • C$\frac14$
   • D$\frac12$

   (b)

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

   [1 mark]

   • A$1$
   • B$\frac32$
   • C$\frac{3}{16}$
   • D$\frac34$

   (c)

   The median of $X$ is $m$. Show that $m^3-12m+8=0$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   The continuous random variable $X$ has cumulative distribution function $\mathrm{F}(x)=0$ for $x<0$, $\mathrm{F}(x)=\frac{x^2+x}{12}$ for $0\le x\le3$, and $\mathrm{F}(x)=1$ for $x>3$.

   (a)

   Find $\mathrm{P}(1<X<2)$.

   [1 mark]

   • A$\frac12$
   • B$\frac13$
   • C$\frac{5}{12}$
   • D$\frac16$

   (b)

   Which expression is the probability density function $f(x)$ for $0\le x\le3$?

   [1 mark]

   • A$2x+1$
   • B$\frac{2x}{12}$
   • C$\frac{2x+1}{12}$
   • D$\frac{x^3}{36}+\frac{x^2}{24}$

   (c)

   Find the lower quartile of $X$ to 3 significant figures.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   The continuous random variable $X$ has probability density function $f(x)=kx^3$ for $0\le x\le2$, and $f(x)=0$ otherwise, where $k$ is a constant.

   (a)

   Find the value of $k$ and the value of $\mathrm{P}(X>1)$.

   [3 marks]

   (b)

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

   [4 marks]

   Total for question 7: 7 marks

8. 8
   The continuous random variable $X$ has cumulative distribution function $\mathrm{F}(x)=0$ for $x<1$, $\mathrm{F}(x)=\frac{x^3-1}{26}$ for $1\le x\le3$, and $\mathrm{F}(x)=1$ for $x>3$.

   (a)

   Find the median and the interquartile range of $X$.

   [6 marks]

   (b)

   Find $\mathrm{E}(X)$ and $\mathrm{Var}(X)$. Using your median from part (a), and the fact that the mode is $3$, state whether the distribution is positively or negatively skewed, giving a reason.

   [6 marks]

   Total for question 8: 12 marks

End of questions