5.03b Solve problems: using pdf

7 Engineering work on the railway network causes an increase in the journey time of commuters travelling into work each morning. The increase in journey time, \(T\) hours, is modelled by a continuous random variable with probability density function $$\mathrm { f } ( t ) = \begin{cases} 4 t \left( 1 - t ^ { 2 } \right) & 0 \leqslant t \leqslant 1 \\ 0 & \text { otherwise } \end{cases}$$

1. Show that \(\mathrm { E } ( T ) = \frac { 8 } { 15 }\).
2. 1. Find the cumulative distribution function, \(\mathrm { F } ( t )\), for \(0 \leqslant t \leqslant 1\).
   2. Hence, or otherwise, for a commuter selected at random, find $$\mathrm { P } ( \text { mean } < T < \text { median } )$$

6 The waiting time, \(T\) minutes, before being served at a local newsagents can be modelled by a continuous random variable with probability density function $$\mathrm { f } ( t ) = \begin{cases} \frac { 3 } { 8 } t ^ { 2 } & 0 \leqslant t < 1 \\ \frac { 1 } { 16 } ( t + 5 ) & 1 \leqslant t \leqslant 3 \\ 0 & \text { otherwise } \end{cases}$$

1. Sketch the graph of f.
2. For a customer selected at random, calculate \(\mathrm { P } ( T \geqslant 1 )\).
3. 1. Show that the cumulative distribution function for \(1 \leqslant t \leqslant 3\) is given by $$\mathrm { F } ( t ) = \frac { 1 } { 32 } \left( t ^ { 2 } + 10 t - 7 \right)$$
   2. Hence find the median waiting time.

8 The continuous random variable \(X\) has the cumulative distribution function $$\mathrm { F } ( x ) = \left\{ \begin{array} { c c } 0 & x \leqslant - 4 \\ \frac { x + 4 } { 9 } & - 4 \leqslant x \leqslant 5 \\ 1 & x \geqslant 5 \end{array} \right.$$

1. Determine the probability density function, \(\mathrm { f } ( x )\), of \(X\).
2. Sketch the graph of f .
3. Determine \(\mathrm { P } ( X > 2 )\).
4. Evaluate the mean and variance of \(X\).

3

1. The continuous random variable \(T\) follows a rectangular distribution with probability density function given by $$\mathrm { f } ( t ) = \left\{ \begin{array} { l c } k & - a \leqslant t \leqslant b \\ 0 & \text { otherwise } \end{array} \right.$$
   1. Express \(k\) in terms of \(a\) and \(b\).
   2. Prove, using integration, that \(\mathrm { E } ( T ) = \frac { 1 } { 2 } ( b - a )\).
2. The error, in minutes, made by a commuter when estimating the journey time by train into London may be modelled by the random variable \(T\) with probability density function $$\mathrm { f } ( t ) = \left\{ \begin{array} { c c } \frac { 1 } { 10 } & - 4 \leqslant t \leqslant 6 \\ 0 & \text { otherwise } \end{array} \right.$$
   1. Write down the value of \(\mathrm { E } ( T )\).
   2. Calculate \(\mathrm { P } ( T < - 3\) or \(T > 3 )\).

2 The error, in minutes, made by Paul in estimating the time that he takes to complete a college assignment may be modelled by the random variable \(T\) with probability density function $$f ( t ) = \left\{ \begin{array} { c c } \frac { 1 } { 30 } & - 5 \leqslant t \leqslant 25 \\ 0 & \text { otherwise } \end{array} \right.$$

1. Find:
   1. \(\mathrm { E } ( T )\);
      (1 mark)
   2. \(\quad \operatorname { Var } ( T )\).
2. Calculate the probability that Paul will make an error of magnitude at least 2 minutes when estimating the time that he takes to complete a given assignment.

8 The continuous random variable \(X\) has probability density function given by $$\mathrm { f } ( x ) = \left\{ \begin{array} { c c } \frac { 1 } { 2 } \left( x ^ { 2 } + 1 \right) & 0 \leqslant x \leqslant 1 \\ ( x - 2 ) ^ { 2 } & 1 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{array} \right.$$

1. Sketch the graph of f.
2. Calculate \(\mathrm { P } ( X \leqslant 1 )\).
3. Show that \(\mathrm { E } \left( X ^ { 2 } \right) = \frac { 4 } { 5 }\).
4. 1. Given that \(\mathrm { E } ( X ) = \frac { 19 } { 24 }\) and that \(\operatorname { Var } ( X ) = \frac { 499 } { k }\), find the numerical value of \(k\).
   2. Find \(\mathrm { E } \left( 5 X ^ { 2 } + 24 X - 3 \right)\).
   3. Find \(\operatorname { Var } ( 12 X - 5 )\).

6 The continuous random variable \(X\) has probability density function defined by $$\mathrm { f } ( x ) = \begin{cases} \frac { 3 } { 8 } x ^ { 2 } & 0 \leqslant x \leqslant \frac { 1 } { 2 } \\ \frac { 3 } { 32 } & \frac { 1 } { 2 } \leqslant x \leqslant 11 \\ 0 & \text { otherwise } \end{cases}$$

1. Sketch the graph of f.
2. Show that:
   1. \(\quad \mathrm { P } \left( X \geqslant 8 \frac { 1 } { 3 } \right) = \frac { 1 } { 4 }\);
   2. \(\quad \mathrm { P } ( X \geqslant 3 ) = \frac { 3 } { 4 }\).
3. Hence write down the exact value of:
   1. the interquartile range of \(X\);
   2. the median, \(m\), of \(X\).
4. Find the exact value of \(\mathrm { P } ( X < m \mid X \geqslant 3 )\).

1 Josephine accurately measures the widths of A4 sheets of paper and then rounds the widths to the nearest 0.1 cm . The rounding error, \(X\) centimetres, follows a rectangular distribution. A randomly selected A4 sheet of paper is measured to be 21.1 cm in width.

1. Write down the limits between which the true width of this A4 sheet of paper lies.
   (1 mark)
2. Write down the value of \(\mathrm { E } ( X )\) and determine the exact value of the standard deviation of \(X\).
3. Calculate \(\mathrm { P } ( - 0.01 \leqslant X \leqslant 0.03 )\).

6 The random variable \(X\) has probability density function defined by $$f ( x ) = \begin{cases} \frac { 1 } { 40 } ( x + 7 ) & 1 \leqslant x \leqslant 5 \\ 0 & \text { otherwise } \end{cases}$$

1. Sketch the graph of f.
2. Find the exact value of \(\mathrm { E } ( X )\).
3. Prove that the distribution function F , for \(1 \leqslant x \leqslant 5\), is defined by $$\mathrm { F } ( x ) = \frac { 1 } { 80 } ( x + 15 ) ( x - 1 )$$
4. Hence, or otherwise:
   1. find \(\mathrm { P } ( 2.5 \leqslant X \leqslant 4.5 )\);
   2. show that the median, \(m\), of \(X\) satisfies the equation \(m ^ { 2 } + 14 m - 55 = 0\).
5. Calculate the value of the median of \(X\), giving your answer to three decimal places.

4 A continuous random variable \(X\) has probability density function defined by $$f ( x ) = \begin{cases} k x ^ { 2 } & 0 \leqslant x \leqslant 3 \\ 9 k & 3 \leqslant x \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$

1. Sketch the graph of f.
2. Show that the value of \(k\) is \(\frac { 1 } { 18 }\).
3. 1. Write down the median value of \(X\).
   2. Calculate the value of the lower quartile of \(X\).

6 The time, in weeks, that a patient must wait to be given an appointment in Holmsoon Hospital may be modelled by a random variable \(T\) having the cumulative distribution function $$\mathrm { F } ( t ) = \begin{cases} 0 & t < 0 \\ \frac { t ^ { 3 } } { 216 } & 0 \leqslant t \leqslant 6 \\ 1 & t > 6 \end{cases}$$

1. Find, to the nearest day, the time within which 90 per cent of patients will have been given an appointment.
2. Find the probability density function of \(T\) for all values of \(t\).
3. Calculate the mean and the variance of \(T\).
4. Calculate the probability that the time that a patient must wait to be given an appointment is more than one standard deviation above the mean.

4 The error, \(X\) millimetres, made when the heights of prospective members of a new gym club are measured can be modelled by a rectangular distribution with the following probability density function. $$f ( x ) = \begin{cases} k & - 4 \leqslant x \leqslant 6 \\ 0 & \text { otherwise } \end{cases}$$

1. State the value of \(k\).
2. Write down the value of \(\mathrm { E } ( X )\).
3. Calculate \(\mathrm { P } ( X > 0 )\).
4. The height of a randomly selected prospective member is measured. Find the probability that the magnitude of the error made exceeds 3.5 millimetres.

7 The time, \(T\) hours, that the supporters of Bracken Football Club have to queue in order to obtain their Cup Final tickets has the following probability density function. $$\mathrm { f } ( t ) = \begin{cases} \frac { 1 } { 5 } & 0 \leqslant t < 3 \\ \frac { 1 } { 45 } t ( 6 - t ) & 3 \leqslant t \leqslant 6 \\ 0 & \text { otherwise } \end{cases}$$

1. Sketch the graph of f.
2. Write down the value of \(\mathrm { P } ( T = 3 )\).
3. Find the probability that a randomly selected supporter has to queue for at least 3 hours in order to obtain tickets.
4. Show that the median queuing time is 2.5 hours.
5. Calculate P (median \(< T <\) mean).

5

1. The continuous random variable \(X\) follows a rectangular distribution with probability density function defined by $$f ( x ) = \begin{cases} \frac { 1 } { b } & 0 \leqslant x \leqslant b \\ 0 & \text { otherwise } \end{cases}$$
   1. Write down \(\mathrm { E } ( X )\).
   2. Prove, using integration, that $$\operatorname { Var } ( X ) = \frac { 1 } { 12 } b ^ { 2 }$$
2. At an athletics meeting, the error, in seconds, made in recording the time taken to complete the 10000 metres race may be modelled by the random variable \(T\), having the probability density function $$f ( t ) = \left\{ \begin{array} { c c } 5 & - 0.1 \leqslant t \leqslant 0.1 \\ 0 & \text { otherwise } \end{array} \right.$$ Calculate \(\mathrm { P } ( | T | > 0.02 )\).

7 The continuous random variable \(X\) has probability density function defined by $$f ( x ) = \begin{cases} \frac { 1 } { 5 } ( 2 x + 1 ) & 0 \leqslant x \leqslant 1 \\ \frac { 1 } { 15 } ( 4 - x ) ^ { 2 } & 1 < x \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$

1. Sketch the graph of f.
2. 1. Show that the cumulative distribution function, \(\mathrm { F } ( x )\), for \(0 \leqslant x \leqslant 1\) is $$\mathrm { F } ( x ) = \frac { 1 } { 5 } x ( x + 1 )$$
   2. Hence write down the value of \(\mathrm { P } ( X \leqslant 1 )\).
   3. Find the value of \(x\) for which \(\mathrm { P } ( X \geqslant x ) = \frac { 17 } { 20 }\).
   4. Find the lower quartile of the distribution.

4 The delay, in hours, of certain flights from Australia may be modelled by the continuous random variable \(T\), with probability density function $$\mathrm { f } ( t ) = \left\{ \begin{array} { c c } \frac { 2 } { 15 } t & 0 \leqslant t \leqslant 3 \\ 1 - \frac { 1 } { 5 } t & 3 \leqslant t \leqslant 5 \\ 0 & \text { otherwise } \end{array} \right.$$

1. Sketch the graph of f.
2. Calculate:
   1. \(\mathrm { P } ( T \leqslant 2 )\);
   2. \(\mathrm { P } ( 2 < T < 4 )\).
3. Determine \(\mathrm { E } ( T )\).

2

1. The continuous random variable \(X\) has a rectangular distribution defined by the probability density function $$f ( x ) = \begin{cases} 0.01 \pi & u \leqslant x \leqslant 11 u \\ 0 & \text { otherwise } \end{cases}$$ where \(u\) is a constant.
   1. Show that \(u = \frac { 10 } { \pi }\).
   2. Using the formulae for the mean and the variance of a rectangular distribution, find, in terms of \(\pi\), values for \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
   3. Calculate exact values for the mean and the variance of the circumferences of circles having diameters of length \(\left( X + \frac { 10 } { \pi } \right)\).
2. A machine produces circular discs which have an area of \(Y \mathrm {~cm} ^ { 2 }\). The distribution of \(Y\) has mean \(\mu\) and variance 25 . A random sample of 100 such discs is selected. The mean area of the discs in this sample is calculated to be \(40.5 \mathrm {~cm} ^ { 2 }\). Calculate a 95\% confidence interval for \(\mu\). Emily believed that the performances of 16-year-old students in their GCSEs are associated with the schools that they attend. To investigate her belief, Emily collected data on the GCSE results for 2010 from four schools in her area. The table shows Emily's collected data, denoted by \(O _ { i }\), together with the corresponding expected frequencies, \(E _ { i }\), necessary for a \(\chi ^ { 2 }\) test.

   \multirow{2}{*}{}	\(\boldsymbol { \geq } \mathbf { 5 }\) GCSEs		\(\mathbf { 1 } \boldsymbol { \leqslant }\) GCSEs < \(\mathbf { 5 }\)		No GCSEs
   	\(O _ { i }\)	\(E _ { i }\)	\(O _ { i }\)	\(E _ { i }\)	\(O _ { i }\)	\(E _ { i }\)
   Jolliffe College for the Arts	187	193.15	93	90.62	30	26.23
   Volpe Science Academy	175	184.43	97	86.52	24	25.05
   Radok Music School	183	183.81	78	86.23	34	24.96
   Bailey Language School	265	248.61	112	116.63	22	33.76

   Emily used these values to correctly conduct a \(\chi ^ { 2 }\) test at the \(1 \%\) level of significance.

7 A continuous random variable \(X\) has probability density function defined by $$f ( x ) = \begin{cases} \frac { 1 } { 6 } ( 4 - x ) & 1 \leqslant x \leqslant 3 \\ \frac { 1 } { 6 } & 3 \leqslant x \leqslant 5 \\ 0 & \text { otherwise } \end{cases}$$

1. Draw the graph of f on the grid on page 6 .
2. Prove that the mean of \(X\) is \(2 \frac { 5 } { 9 }\).
3. Calculate the exact value of:
   1. \(\mathrm { P } ( X > 2.5 )\);
   2. \(\mathrm { P } ( 1.5 < X < 4.5 )\);
   3. \(\mathrm { P } ( X > 2.5\) and \(1.5 < X < 4.5 )\);
   4. \(\mathrm { P } ( X > 2.5 \mid 1.5 < X < 4.5 )\). \includegraphics[max width=\textwidth, alt={}, center]{bc21c177-6cd8-4c79-8782-d17f0238ce17-6_1340_1363_317_383}

3 Mehreen lives a 2-minute walk away from a tram stop. Trams run every 10 minutes into the city centre, taking 20 minutes to get there. Every morning, Mehreen leaves her house, walks to the tram stop and catches the first tram that arrives. When she arrives at the city centre, she then has a 5-minute walk to her office. The total time, \(T\) minutes, for Mehreen's journey from house to office may be modelled by a rectangular distribution with probability density function $$\mathrm { f } ( t ) = \begin{cases} 0.1 & a \leqslant t \leqslant b \\ 0 & \text { otherwise } \end{cases}$$

1. 1. Explain why \(a = 27\).
   2. State the value of \(b\).
2. Find the values of \(\mathrm { E } ( T )\) and \(\operatorname { Var } ( T )\).
3. Find the probability that the time for Mehreen's journey is within 5 minutes of half an hour.

7 A continuous random variable \(X\) has the probability density function defined by $$\mathrm { f } ( x ) = \left\{ \begin{array} { c c } x ^ { 2 } & 0 \leqslant x \leqslant 1 \\ \frac { 1 } { 3 } ( 5 - 2 x ) & 1 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{array} \right.$$

1. Sketch the graph of f on the axes below.
2. 1. Find the cumulative distribution function, F , for \(0 \leqslant x \leqslant 1\).
   2. Hence, or otherwise, find the value of the lower quartile of \(X\).
3. 1. Show that the cumulative distribution function for \(1 \leqslant x \leqslant 2\) is defined by $$\mathrm { F } ( x ) = \frac { 1 } { 3 } \left( 5 x - x ^ { 2 } - 3 \right)$$
   2. Hence, or otherwise, find the value of the upper quartile of \(X\). \includegraphics[max width=\textwidth, alt={}, center]{03c1e107-3377-4b0d-9daf-7f70233c18b5-5_554_1050_1217_424}

4 A continuous random variable \(X\) has a probability density function defined by $$f ( x ) = \begin{cases} \frac { 1 } { k } & a \leqslant x \leqslant b \\ 0 & \text { otherwise } \end{cases}$$ where \(b > a > 0\).

1. 1. Prove that \(k = b - a\).
   2. Write down the value of \(\mathrm { E } ( X )\).
   3. Show, by integration, that \(\mathrm { E } \left( X ^ { 2 } \right) = \frac { 1 } { 3 } \left( b ^ { 2 } + a b + a ^ { 2 } \right)\).
   4. Hence derive a simplified formula for \(\operatorname { Var } ( X )\).
2. Given that \(a = 4\) and \(\operatorname { Var } ( X ) = 3\), find the numerical value of \(\mathrm { E } ( X )\).
   \includegraphics[max width=\textwidth, alt={}]{34517557-011e-4956-be7d-b26fe5e64d0a-08_1347_1707_1356_153}

2 The continuous random variable \(X\) has probability density function defined by $$f ( x ) = \begin{cases} \frac { 1 } { k } & a \leqslant x \leqslant b \\ 0 & \text { otherwise } \end{cases}$$

1. Write down, in terms of \(a\) and \(b\), the value of \(k\).
2. 1. Given that \(\mathrm { E } ( X ) = 1\) and \(\operatorname { Var } ( X ) = 3\), find the values of \(a\) and \(b\).
   2. Four independent values of \(X\) are taken. Find the probability that exactly one of these four values is negative.
      [0pt] [3 marks]

6 The continuous random variable \(X\) has the cumulative distribution function $$\mathrm { F } ( x ) = \begin{cases} 0 & x < 0 \\ \frac { 1 } { 2 } x - \frac { 1 } { 16 } x ^ { 2 } & 0 \leqslant x \leqslant 4 \\ 1 & x > 4 \end{cases}$$

1. Find the probability that \(X\) lies between 0.4 and 0.8 .
2. Show that the probability density function, \(\mathrm { f } ( x )\), of \(X\) is given by $$f ( x ) = \begin{cases} \frac { 1 } { 2 } - \frac { 1 } { 8 } x & 0 \leqslant x \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$
3. 1. Find the value of \(\mathrm { E } ( X )\).
   2. Show that \(\operatorname { Var } ( X ) = \frac { 8 } { 9 }\).
4. The continuous random variable \(Y\) is defined by $$Y = 3 X - 2$$ Find the values of \(\mathrm { E } ( Y )\) and \(\operatorname { Var } ( Y )\).

4. A continuous random variable \(X\) has the cumulative distribution function $$\mathrm { F } ( x ) = \left\{ \begin{array} { l } 0 \\ \frac { 1 } { 84 } \left( x ^ { 2 } - 16 \right) \\ 1 \end{array} \right.$$ $$\begin{aligned} & x < 4 , \\ & 4 \leq x \leq 10 , \\ & x > 10 . \end{aligned}$$

1. Find the median value of \(X\).
2. Find the interquartile range for \(X\).
3. Find the probability density function \(\mathrm { f } ( x )\) of \(X\).
4. Sketch the graph of \(\mathrm { f } ( x )\) and hence write down the mode of \(X\), explaining how you obtain your answer from the graph. \section*{STATISTICS 2 (A) TEST PAPER 1 Page 2}

6. Patients suffering from 'flu are treated with a drug. The number of days, \(t\), that it then takes for them to recover is modelled by the continuous random variable \(T\) with the probability density function $$\begin{array} { l l } \mathrm { f } ( t ) = \frac { 3 t ^ { 2 } ( 4 - t ) } { 64 } & 0 \leq t \leq 4 \\ \mathrm { f } ( t ) = 0 & \text { otherwise. } \end{array}$$

1. Find the mean and standard deviation of \(T\).
2. Find the probability that a patient takes more than 3 days to recover.
3. Two patients are selected at random. Find the probability that they both recover within three days.
4. Comment on the suitability of the model.