5.03a Continuous random variables: pdf and cdf

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.

6. The continuous random variable \(X\) has a probability density function $$\mathrm { f } ( x ) = \left\{ \begin{array} { c c } k ( x - 2 ) & 2 \leqslant x \leqslant 3 \\ k & 3 < x < 5 \\ k ( 6 - x ) & 5 \leqslant x \leqslant 6 \\ 0 & \text { otherwise } \end{array} \right.$$ where \(k\) is a positive constant.

1. Sketch the graph of \(\mathrm { f } ( x )\).
2. Show that the value of \(k\) is \(\frac { 1 } { 3 }\)
3. Define fully the cumulative distribution function \(\mathrm { F } ( x )\).
4. Hence find the 90th percentile of the distribution.
5. Find \(\mathrm { P } [ \mathrm { E } ( X ) < X < 5.5 ]\)

   Leave
   blank
   END

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.

1. David aims to catch the train to work each morning. The scheduled departure time of the train is 0830

The number of minutes after 0830 that the train departs may be modelled by the random variable \(X\). Given that \(X\) has a continuous uniform distribution over \([ \alpha , \beta ]\) and that \(\mathrm { E } ( X ) = 4\) and \(\operatorname { Var } ( X ) = 12\)

1. find the value of \(\alpha\) and the value of \(\beta\). Each morning, the probability that David oversleeps is 0.05 If David oversleeps he will be late for work. If he does not oversleep he will be in time to catch the train, but will be late for work if the train departs after 0835
2. Find the probability that David will be late for work. Given that David is late for work,
3. find the probability that he overslept.
   

1. The continuous random variable \(X\) has the following cumulative distribution function

$$\mathrm { F } ( x ) = \left\{ \begin{array} { c c } 0 & x \leqslant 1 \\ \frac { 4 } { 15 } ( x - 1 ) & 1 < x \leqslant 2 \\ k \left( \frac { a x ^ { 3 } } { 3 } - \frac { x ^ { 4 } } { 4 } \right) + b & 2 < x \leqslant 4 \\ 1 & x > 4 \end{array} \right.$$ where \(k , a\) and \(b\) are constants.
Given that the mode of \(X\) is \(\frac { 8 } { 3 }\)

1. show that \(a = 4\)
2. Find \(\mathrm { P } ( X < 2.5 )\) giving your answer to 3 significant figures.
   

1. The lifetime, in tens of hours, of a certain delicate electrical component can be modelled by the random variable \(X\) with probability density function

$$f ( x ) = \begin{cases} \frac { 1 } { 42 } x , & 0 \leq x < 6 \\ \frac { 1 } { 7 } & 6 \leq x \leq 10 \\ 0 , & \text { otherwise } \end{cases}$$

1. Sketch \(\mathrm { f } ( x )\) for all values of \(x\).
2. Find the probability that a component lasts at least 50 hours. A particular device requires two of these components and it will not operate if one or more of the components fail. The device has just been fitted with two new components and the lifetimes of these two components are independent.
3. Find the probability that the device breaks down within the next 50 hours.
   

2. The continuous random variable \(X\) represents the error, in mm, made when a machine cuts piping to a target length. The distribution of \(X\) is rectangular over the interval \([ - 5.0,5.0 ]\). Find

1. \(\mathrm { P } ( X < - 4.2 )\),
2. \(\mathrm { P } ( | X | < 1.5 )\). A supervisor checks a random sample of 10 lengths of piping cut by the machine.
3. Find the probability that more than half of them are within 1.5 cm of the target length.
   (3 marks)
   If \(X < - 4.2\), the length of piping cannot be used. At the end of each day the supervisor checks a random sample of 60 lengths of piping.
4. Use a suitable approximation to estimate the probability that no more than 2 of these lengths of piping cannot be used.
   (5 marks)

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.

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

1. Show that \(\bar { X }\) is a biased estimator of \(\alpha\), and state the bias. Given that \(Y = k \bar { X }\) is an unbiased estimator for \(\alpha\)
2. find the value of \(k\). A random sample of 10 values of \(X\) is taken and the results are as follows $$\begin{array} { l l l l l l l l l l } 3 & 5 & 8 & 12 & 4 & 13 & 10 & 8 & 5 & 12 \end{array}$$
3. Hence estimate the maximum value of \(X\)

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}

Navtej travels to work by train. A train leaves the station every 7 minutes and Navtej's arrival at the station is independent of when the train is due to leave.

1. Write down a suitable model for the distribution of the time, \(T\) minutes, that he has to wait for a train to leave.
2. Find the mean and standard deviation of \(T\) During a 10-week period, Navtej travels to work by train on 46 occasions.
3. Estimate the probability that the mean length of time that he has to wait for a train to leave is between 3.4 and 3.6 minutes.
4. State a necessary assumption for the calculation in part (c).

5. The discrete random variable \(Y\) has the following cumulative distribution function.

1. Write down the probability distribution of \(Y\).
2. Find \(\mathrm { P } ( 1 \leq Y < 3 )\).
3. Show that \(\mathrm { E } ( Y ) = 2.7\)
4. Find \(\mathrm { E } ( 2 Y + 4 )\).
5. Find \(\operatorname { Var } ( Y )\).

1. The discrete random variable \(Y\) has the following probability distribution.

Find

1. \(\mathrm { F } ( 0.5 )\),
2. \(\mathrm { P } \left( { } ^ { - } 1 < Y < 1.9 \right)\),
3. \(\mathrm { E } ( Y )\),
4. \(\mathrm { E } ( 3 Y - 1 )\).
   

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

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 \(T > 3 )\).

7 The waiting time, \(X\) minutes, for fans to gain entrance to see an event may be modelled by a continuous random variable having the distribution function defined by $$\mathrm { F } ( x ) = \begin{cases} 0 & x < 0 \\ \frac { 1 } { 2 } x & 0 \leqslant x \leqslant 1 \\ \frac { 1 } { 54 } \left( x ^ { 3 } - 12 x ^ { 2 } + 48 x - 10 \right) & 1 \leqslant x \leqslant 4 \\ 1 & x > 4 \end{cases}$$

1. 1. Sketch the graph of F.
   2. Explain why the value of \(q _ { 1 }\), the lower quartile of \(X\), is \(\frac { 1 } { 2 }\).
   3. Show that the upper quartile, \(q _ { 3 }\), satisfies \(1.6 < q _ { 3 } < 1.7\).
2. The probability density function of \(X\) is defined by $$\mathrm { f } ( x ) = \begin{cases} \alpha & 0 \leqslant x \leqslant 1 \\ \beta ( x - 4 ) ^ { 2 } & 1 \leqslant x \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$
   1. Show that the exact values of \(\alpha\) and \(\beta\) are \(\frac { 1 } { 2 }\) and \(\frac { 1 } { 18 }\) respectively.
   2. Hence calculate \(\mathrm { E } ( X )\).

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.