Further Statistics 1: Continuous random variablesAQA A-Level Further Maths: Topic test

1. 1
   The thickness, $X$ mm, of the coating on a lens is modelled by a continuous random variable with 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 $k$.

   [1 mark]

   • A$\frac{1}{16}$
   • B$\frac14$
   • C$\frac12$
   • D$\frac13$

   (b)

   Find $P(X>1)$.

   [1 mark]

   • A$\frac{1}{16}$
   • B$\frac{1}{4}$
   • C$\frac{3}{4}$
   • D$\frac{15}{16}$

   (c)

   Find the median thickness.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A parcel consists of a box of mass $U$ kg and its contents of mass $V$ kg, where $U$ and $V$ are independent random variables. $E(U)=2.4$, $\mathrm{Var}(U)=0.04$, $E(V)=15.6$ and $\mathrm{Var}(V)=0.25$.

   (a)

   Find $E(U+V)$.

   [1 mark]

   • A$18.0$
   • B$13.2$
   • C$36.0$
   • D$18.29$

   (b)

   Find $\mathrm{Var}(V-U)$.

   [1 mark]

   • A$0.21$
   • B$0.54$
   • C$0.29$
   • D$13.2$

   (c)

   Find the mean and variance of $D=V-2U$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The time, $X$ hours, that a student spends on homework one evening has cumulative distribution function $F(x)=0$ for $x<1$, $F(x)=\frac{x^2-1}{8}$ for $1\le x\le3$, and $F(x)=1$ for $x>3$.

   (a)

   Find the probability density function $f(x)$ of $X$.

   [3 marks]

   (b)

   Find $E(X)$ and the median of $X$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The duration, $X$ minutes, of a taxi journey across a city is modelled by a continuous rectangular distribution on the interval $[15,35]$. The fare for a journey is $\pounds C$, where $C=2+0.8X$.

   (a)

   Find $E(C)$ and $\mathrm{Var}(C)$.

   [6 marks]

   (b)

   Find the cumulative distribution function $F(x)$ of $X$ for all $x$. Hence find the value of $t$ such that the probability that a journey lasts more than $t$ minutes is $0.1$.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   The deviation, $X$ mm, of a drilled hole from its target position along one axis has probability density function $f(x)=\frac34(1-x^2)$ for $-1\le x\le1$, and $f(x)=0$ otherwise.

   (a)

   Find $E(X)$.

   [1 mark]

   • A$\frac34$
   • B$\frac15$
   • C$0$
   • D$\frac{3}{16}$

   (b)

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

   [1 mark]

   • A$\frac15$
   • B$\frac12$
   • C$\frac{1}{\sqrt5}$
   • D$\frac{1}{25}$

   (c)

   Find $E(10X^2+3)$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   The delay, $X$ days, in the delivery of an online order is modelled as follows. The order arrives on time with probability $0.3$, so $P(X=0)=0.3$. Otherwise the delay lies between $0$ and $2$ days, and for $0<x\leq2$ the function $f(x)=kx$, where $k$ is a constant, is the density of $X$ in the sense that $P(a<X<b)=\int_a^bf(x)\,\mathrm{d}x$ for $0\leq a<b\leq2$.

   (a)

   Find the value of $k$.

   [1 mark]

   • A$0.5$
   • B$0.7$
   • C$0.15$
   • D$0.35$

   (b)

   Find $P(X\le1)$.

   [1 mark]

   • A$0.175$
   • B$0.475$
   • C$0.65$
   • D$0.3$

   (c)

   Find $E(X)$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   The daily outputs, in tonnes, of two independent plants are modelled by random variables $X$ and $Y$, with $E(X)=14$, $\mathrm{Var}(X)=2.25$, $E(Y)=9$ and $\mathrm{Var}(Y)=1.44$. The daily net value, in thousands of pounds, is $W=20+4X-3Y$.

   (a)

   Find $E(W)$ and $\mathrm{Var}(W)$.

   [3 marks]

   (b)

   The outputs of plant $X$ on three different days are independent observations $X_1$, $X_2$ and $X_3$. Find the mean and variance of $X_1+X_2+X_3$ and the variance of $3X$. Explain why the two variances differ.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   The distance, $X$ metres, from the centre of a target at which a thrown ball lands is modelled by a continuous random variable with probability density function $f(x)=\frac{3}{16}(4-x^2)$ for $0\le x\le2$, and $f(x)=0$ otherwise.

   (a)

   Find the cumulative distribution function $F(x)$ for all $x$. Hence find $P(0.5<X<1.5)$.

   [6 marks]

   (b)

   Find $E(X)$ and $\mathrm{Var}(X)$. Two independent throws land at distances $X_1$ and $X_2$. Find the standard deviation of $X_1+X_2$.

   [6 marks]

   Total for question 8: 12 marks

End of questions