5.03c Calculate mean/variance: by integration

The random variable \(R\) has a continuous uniform distribution over the interval [5,9]

1. Specify fully the probability density function of \(R\).
2. Find \(\mathrm { P } ( 7 < R < 10 )\) The random variable \(A\) is the area of a circle radius \(R \mathrm {~cm}\).
3. Find \(\mathrm { E } ( \mathrm { A } )\)

7. The weight, \(X \mathrm {~kg}\), of staples in a bin full of paper has probability density function $$f ( x ) = \left\{ \begin{array} { c c } \frac { 9 x - 3 x ^ { 2 } } { 10 } & 0 \leqslant x < 2 \\ 0 & \text { otherwise } \end{array} \right.$$ Use integration to find

1. \(\mathrm { E } ( X )\)
2. \(\operatorname { Var } ( X )\)
3. \(\mathrm { P } ( X > 1.5 )\) Peter raises money by collecting paper and selling it for recycling. A bin full of paper is sold for \(\pounds 50\) but if the weight of the staples exceeds 1.5 kg it sells for \(\pounds 25\)
4. Find the expected amount of money Peter raises per bin full of paper. Peter could remove all the staples before the paper is sold but the time taken to remove the staples means that Peter will have \(20 \%\) fewer bins full of paper to sell.
5. Decide whether or not Peter should remove all the staples before selling the bins full of paper. Give a reason for your answer.
   \href{http://PhysicsAndMathsTutor.com}{PhysicsAndMathsTutor.com}

3. The lifetime, \(X\), in tens of hours, of a battery is modelled by the probability density function $$f ( x ) = \left\{ \begin{array} { c c } \frac { 1 } { 9 } x ( 4 - x ) & 1 \leqslant x \leqslant 4 \\ 0 & \text { otherwise } \end{array} \right.$$ Use algebraic integration to find

1. \(\mathrm { E } ( X )\)
2. \(\mathrm { P } ( X > 2.5 )\) A radio runs using 2 of these batteries, both of which must be working. Two fully-charged batteries are put into the radio.
3. Find the probability that the radio will be working after 25 hours of use. Given that the radio is working after 16 hours of use,
4. find the probability that the radio will be working after being used for another 9 hours.

4. The continuous random variable \(X\) is uniformly distributed over the interval \([ \alpha , \beta ]\) Given that \(\mathrm { E } ( X ) = 3.5\) and \(\mathrm { P } ( X > 5 ) = \frac { 2 } { 5 }\)

1. find the value of \(\alpha\) and the value of \(\beta\) Given that \(\mathrm { P } ( X < c ) = \frac { 2 } { 3 }\)
2. 1. find the value of \(c\)
   2. find \(\mathrm { P } ( c < X < 9 )\) A rectangle has a perimeter of 200 cm . The length, \(S \mathrm {~cm}\), of one side of this rectangle is uniformly distributed between 30 cm and 80 cm .
3. Find the probability that the length of the shorter side of the rectangle is less than 45 cm .

1. The length of time, \(T\), minutes, spent completing a particular task has probability density function

$$f ( t ) = \left\{ \begin{array} { c c } \frac { 1 } { 2 } ( t - 1 ) & 1 < t \leqslant 2 \\ \frac { 1 } { 16 } \left( 14 t - 3 t ^ { 2 } - 8 \right) & 2 < t \leqslant 4 \\ 0 & \text { otherwise } \end{array} \right.$$

1. Use algebraic integration to find \(\mathrm { E } ( T )\) Given that \(\mathrm { E } \left( T ^ { 2 } \right) = \frac { 267 } { 40 }\)
2. find \(\operatorname { Var } ( T )\)
3. Find the cumulative distribution function \(\mathrm { F } ( t )\)
4. Find the 20th percentile of the time taken to complete the task.
5. Find the probability that the time spent completing the task is more than 1.5 minutes. Given that a person has already spent 1.5 minutes on the task,
6. find the probability that this person takes more than 3 minutes to complete the task.

5. The continuous random variable \(T\) represents the time in hours that students spend on homework. The cumulative distribution function of \(T\) is $$\mathrm { F } ( t ) = \begin{cases} 0 , & t < 0 \\ k \left( 2 t ^ { 3 } - t ^ { 4 } \right) & 0 \leq t \leq 1.5 \\ 1 , & t > 1.5 \end{cases}$$ where \(k\) is a positive constant.

1. Show that \(k = \frac { 16 } { 27 }\).
2. Find the proportion of students who spend more than 1 hour on homework.
3. Find the probability density function \(\mathrm { f } ( t )\) of \(T\).
4. Show that \(\mathrm { E } ( T ) = 0.9\).
5. Show that \(\mathrm { F } ( \mathrm { E } ( T ) ) = 0.4752\). A student is selected at random. Given that the student spent more than the mean amount of time on homework,
6. find the probability that this student spent more than 1 hour on homework.

8. The random variable \(X\) has a continuous uniform distribution over the interval \([ \alpha + 3,2 \alpha + 9 ]\) where \(\alpha\) is a constant. The mean of a random sample of size \(n\), taken from this distribution, is denoted by \(\bar { X }\)

1. Show that \(\bar { X }\) is a biased estimator of \(\alpha\)
2. Hence find the bias, in terms of \(\alpha\), when \(\bar { X }\) is used as an estimator of \(\alpha\) Given that \(Y = \frac { 2 \bar { X } } { 3 } + k\) is an unbiased estimator of \(\alpha\)
3. find the value of the constant \(k\) A random sample of 8 values of \(X\) is taken and the results are as follows
   4.8
   5.8
   6.5
   7.1
   8.2
   9.5
   9.9
   10.6
4. Use the sample to estimate the maximum value that \(X\) can take.
   \includegraphics[max width=\textwidth, alt={}]{3b6aaa8a-aeac-4a44-820b-e82f317d0c85-28_2633_1828_119_121}

4

1. A random variable \(X\) has probability density function defined by $$\mathrm { f } ( x ) = \begin{cases} k & a < x < b \\ 0 & \text { otherwise } \end{cases}$$
   1. Show that \(k = \frac { 1 } { b - a }\).
   2. Prove, using integration, that \(\mathrm { E } ( X ) = \frac { 1 } { 2 } ( a + b )\).
2. The error, \(X\) grams, made when a shopkeeper weighs out loose sweets can be modelled by a rectangular distribution with the following probability density function: $$f ( x ) = \begin{cases} k & - 2 < x < 4 \\ 0 & \text { otherwise } \end{cases}$$
   1. Write down the value of the mean, \(\mu\), of \(X\).
   2. Evaluate the standard deviation, \(\sigma\), of \(X\).
   3. Hence find \(\mathrm { P } \left( X < \frac { 2 - \mu } { \sigma } \right)\).

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 $$\mathrm { P } ( \text { mean } < T < \text { median } )$$

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.

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 \(\mathrm { P } ( | T | > 0.02 )\).

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.

3 The continuous random variable \(X\) has a cumulative distribution function defined by $$\mathrm { F } ( x ) = \left\{ \begin{array} { c l } 0 & x < - 5 \\ \frac { x + 5 } { 20 } & - 5 \leqslant x \leqslant 15 \\ 1 & x > 15 \end{array} \right.$$

1. Show that, for \(- 5 \leqslant x \leqslant 15\), the probability density function, \(\mathrm { f } ( x )\), of \(X\) is given by \(\mathrm { f } ( x ) = \frac { 1 } { 20 }\).
   (1 mark)
2. Find:
   1. \(\mathrm { P } ( X \geqslant 7 )\);
   2. \(\mathrm { P } ( X \neq 7 )\);
   3. \(\mathrm { E } ( X )\);
   4. \(\mathrm { E } \left( 3 X ^ { 2 } \right)\).

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}