5.03a Continuous random variables: pdf and cdf

4 An archer fires arrows at a circular target of radius 50 cm . The distance in cm that an arrow lands from the centre of the target is modelled by the random variable \(X\), with probability density function given by \(f ( x ) = \begin{cases} a x & 0 \leqslant x \leqslant 50 , \\ 0 & \text { otherwise, } \end{cases}\) where \(a\) is a constant.

1. Determine the value of \(a\).
2. Determine the probability that an arrow will land within 5 cm of the centre of the target.
3. Determine the median distance from the centre of the target that an arrow will land.

10 Ben takes an underground train to work and back home each day. The waiting time is defined as the time from when he reaches the station platform until he boards the train. On his way to work the waiting time is \(X\) minutes, where \(X\) is modelled by a continuous uniform distribution on \([ 0,6 ]\). On his way back from work, the waiting time is \(Y\) minutes, where \(Y\) is modelled by a continuous uniform distribution on [0,4]. Ben's total waiting time for both journeys is \(Z\) minutes, where \(Z = X + Y\). You should assume that \(X\) and \(Y\) are independent.

1. Find \(\mathrm { E } ( \mathrm { Z } )\).
2. Ben thinks that \(Z\) will be well modelled by a continuous uniform distribution on \([ 0,10 ]\). By considering variances, show that he is not correct.
3. Ben's friend Jamila constructs the spreadsheet below, which shows a simulation of 20 values of \(X , Y\) and \(Z\). All of the values have been rounded to 2 decimal places.

   \multirow[b]{3}{*}{ 1 2 }	A	B	C
   	X	Y	Z
   	1.17	3.83	5.01
   3	2.01	0.81	2.82
   4	1.27	1.52	2.78
   5	1.41	3.94	5.35
   6	4.11	2.94	7.05
   7	1.76	0.96	2.72
   8	3.29	0.98	4.27
   9	0.77	0.22	0.99
   10	0.99	1.44	2.43
   11	4.79	2.43	7.22
   12	3.82	3.93	7.75
   13	5.25	2.74	7.99
   14	2.64	0.48	3.12
   15	1.54	2.18	3.72
   16	2.71	1.66	4.36
   17	0.04	3.24	3.28
   18	5.95	3.12	9.07
   19	5.22	1.21	6.42
   20	4.16	0.11	4.27
   21	1.02	0.99	2.01
   22

   Write down an estimate of \(\mathrm { P } ( Z > 6 )\).
4. Use a Normal approximation to determine the probability that Ben's total waiting time when travelling to and from work on 40 days is more than 210 minutes.

11 The discrete random variable \(X\) has a uniform distribution over the set of all integers between 25 and \(n\) inclusive, where \(n\) is a positive integer with \(n > 25\).

1. Determine \(\mathrm { P } \left( \mathrm { X } < \frac { \mathrm { n } + 25 } { 2 } \right)\) in each of the following cases.

12 The cumulative distribution function of the continuous random variable \(X\) is given by \(F ( x ) = \begin{cases} 0 & x < 20 , \\ a \left( x ^ { 2 } + b x + c \right) & 20 \leqslant x \leqslant 30 , \\ 1 & x > 30 , \end{cases}\) where \(a\), \(b\) and \(c\) are constants.
You are given that \(\mathrm { P } ( X < 25 ) = \frac { 11 } { 24 }\).

1. Find \(\mathrm { P } ( X > 27 )\).
2. Find the 90th percentile of \(X\).

11 The length of time in minutes for which a particular geyser erupts is modelled by the continuous random variable \(T\) with cumulative distribution function given by \(\mathrm { F } ( t ) = \begin{cases} 0 & t \leqslant 2 , \\ k \left( 8 t ^ { 2 } - t ^ { 3 } - 24 \right) & 2 < t < 4 , \\ 1 & t \geqslant 4 , \end{cases}\) where \(k\) is a positive constant.

1. Show that \(k = \frac { 1 } { 40 }\).
2. Find the probability that a randomly selected eruption time lies between 2.5 and 3.5 minutes.
3. Show that the median \(m\) of the distribution satisfies the equation \(m ^ { 3 } - 8 m ^ { 2 } + 44 = 0\).
4. Verify that the median eruption time is 2.95 minutes, correct to 2 decimal places. The mean and standard deviation of \(T\) are denoted by \(\mu\) and \(\sigma\) respectively.
5. Find \(\mathrm { P } ( \mu - \sigma < T < \mu + \sigma )\).
6. Sketch the graph of the probability density function of \(T\).
7. A Normally distributed random variable \(X\) has the same mean and standard deviation as \(T\). By considering the shape of the Normal distribution, and without doing any calculations, explain whether \(\mathrm { P } ( \mu - \sigma < X < \mu + \sigma )\) will be greater than, equal to or less than the probability that you calculated in part (e).

10 Sarah takes a bus to work each weekday morning and returns each evening. The times in minutes that she has to wait for the bus in the morning and evening are modelled by uniform distributions over the intervals \([ 0,10 ]\) and \([ 0,6 ]\) respectively. The times in minutes for the bus journeys in the morning and evening are modelled by \(\mathrm { N } ( 25,4 )\) and \(\mathrm { N } ( 28,16 )\) respectively. You should assume that all of the times are independent. The total time in minutes that she takes for her two journeys, including the waiting times, is denoted by the random variable \(T\). The spreadsheet below shows the first 20 rows of a simulation of 500 return journeys. It also shows in column H the numbers of values of \(T\) that are less than or equal to the corresponding values in column G. For example, there are 156 out of the 500 simulated values of \(T\) which are less than or equal to 58 minutes. All of the times have been rounded to 2 decimal places.

1. Use the spreadsheet output to estimate each of the following.

11 The continuous random variable \(X\) has probability density function given by \(f ( x ) = \begin{cases} a x ^ { 2 } & 0 \leqslant x < 2 , \\ b ( 3 - x ) ^ { 2 } & 2 \leqslant x \leqslant 3 , \\ 0 & \text { otherwise } \end{cases}\) where \(a\) and \(b\) are positive constants.

1. Given that \(\mathrm { E } ( X ) = 2\), determine the values of \(a\) and \(b\).
2. Determine the median value of \(X\).
3. A random sample of 50 observations of \(X\) is selected. Given that \(\operatorname { Var } ( X ) = 0.2\), determine an estimate of the probability that the mean value of the 50 observations is less than 1.9.

2. The smallest angle \(\theta\), in degrees, of a right-angled triangle with hypotenuse 8 cm , is uniformly distributed across all possible values. \includegraphics[max width=\textwidth, alt={}, center]{8f47b2ff-f954-42ec-8ecc-fc64313a7b89-04_419_696_479_687}

1. Find the mean and standard deviation of \(\theta\).
2. The shortest side of the triangle is of length \(X \mathrm {~cm}\). Find the probability that \(X\) is greater than 5 .

3. The number of claims made to the home insurance department of an insurance company follows a Poisson distribution with mean 4 per day.

1. Find the probability that more than 11 claims are made in a 2 -day period. The number of claims made in a day to the pet insurance department of the same company follows a Poisson distribution with parameter \(\lambda\). An insurance company worker notices that the probability of two claims being made in a day is three times the probability of four claims being made in a day.
2. Determine the value of \(\lambda\). The car insurance department models the length of time between claims for drivers aged 17 to 21 as an exponential distribution with mean 10 months. Rachel is 17 years old and has just passed her test. Her father says he will give her the car that they share if she does not make a claim in the first 12 months.
3. What is the probability that her father gives her the car?

4. The continuous random variable, \(X\), has the following probability density function $$f ( x ) = \begin{cases} k x & \text { for } 0 \leqslant x < 1 \\ k x ^ { 3 } & \text { for } 1 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant. \includegraphics[max width=\textwidth, alt={}, center]{4ecf99c5-c4b3-41b7-a8df-a7c2ca7fcd6a-3_851_935_826_678}

1. Show that \(k = \frac { 4 } { 17 }\).
2. Determine \(\mathrm { E } ( X )\).
3. Calculate \(\mathrm { E } ( 3 X - 1 )\) and \(\operatorname { Var } ( 3 X - 1 )\).

3. Two basketball players, Steph and Klay, score baskets at random at a rate of \(2 \cdot 1\) and \(1 \cdot 9\) respectively per quarter of a game. Assume that baskets are scored independently, and that Steph and Klay each play all four quarters of the game.

1. Stating the model that you are using, find the probability that they will score a combined total of exactly 20 baskets in a randomly selected game.
2. A quarter of a game lasts 12 minutes.
   1. State the distribution of the time between baskets for Steph. Give the mean and standard deviation of this distribution.
   2. Given that Klay scores at the end of the third minute in a quarter of a game, find the probability that Klay doesn't score for the rest of the quarter.
3. When practising, Klay misses \(4 \%\) of the free throws he takes. One week he takes 530 free throws. Calculate the probability that he misses more than 25 free throws.

4. The continuous random variable \(R\) has probability density function \(f ( r )\) given by $$f ( r ) = \begin{cases} k r ( b - r ) & \text { for } 1 \leqslant r \leqslant 4 , \\ 0 & \text { otherwise } , \end{cases}$$ where \(k\) and \(b\) are positive constants.

1. Explain why \(b \geqslant 4\).
2. Given that \(b = 4\),
   1. show that \(k = \frac { 1 } { 9 }\),
   2. find an expression for \(F ( r )\), valid for \(1 \leqslant r \leqslant 4\), where \(F\) denotes the cumulative distribution function of \(R\),
   3. find the probability that \(R\) lies between 2 and 3 .

Dave and Llinos like to go fishing. When they go fishing, on average, Dave catches 4.3 fish per day and Llinos catches 3.8 fish per day. A day of fishing is assumed to be 8 hours.

1. 1. Calculate the probability that they will catch fewer than 2 fish in total on a randomly selected half-day of fishing.
   2. Justify any distribution you have used in answering (a)(i).
      
   (b) On a randomly selected day, Dave starts fishing at 7 am. Given that Dave has not caught a fish by 11 am,
   1. find the expected time he catches his first fish,
   2. calculate the probability that he will not catch a fish by 3 pm .
      
   (c) On average, only \(2 \%\) of the fish that Llinos catches are trout. Over the course of a year, she catches 950 fish. Calculate the probability that at least 30 of these fish are trout. [3]
   [0pt] she catches 950 fish. Calculate the probability that at least 30 of these fish are trout. [3]
2. State, with a reason, a distribution, including any parameters, that could approximate the distribution used in part (c).
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2 The continuous random variable \(Y\) has cumulative distribution function defined by $$\mathrm { F } ( y ) = \left\{ \begin{array} { c c } 0 & y < 0 \\ \frac { y ^ { 2 } } { 36 } & 0 \leq y \leq 6 \\ 1 & y > 6 \end{array} \right.$$ Find the value of \(\mathrm { P } ( Y > 4 )\) Circle your answer. \(\frac { 4 } { 9 }\) \(\frac { 5 } { 9 }\) \(\frac { 16 } { 27 }\) \(\frac { 11 } { 27 }\)

1. The continuous random variable X has probability density function

$$f ( x ) = \begin{cases} \frac { 1 } { 8 } & 1 \leqslant x \leqslant 9 \\ 0 & \text { otherwise } \end{cases}$$

1. Write down the name given to this distribution. The continuous random variable \(Y = 5 - 2 X\)
2. Find \(\mathrm { P } ( Y > 0 )\)
3. Find \(\mathrm { E } ( Y )\)
4. Find \(\mathrm { P } ( Y < 0 \mid X < 7.5 )\)

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

$$\mathrm { F } ( x ) = \left\{ \begin{array} { l r } 0 & x < 3 \\ c - 4.5 x ^ { n } & 3 \leqslant x \leqslant 9 \\ 1 & x > 9 \end{array} \right.$$ where \(c\) is a positive constant and \(n\) is an integer.

1. Showing all stages of your working, find the value of \(c\) and the value of \(n\)
2. Find the lower quartile of \(X\)

1. Lloyd regularly takes a break from work to go to the local cafe. The amount of time Lloyd waits to be served, in minutes, is modelled by the continuous random variable \(T\), having probability density function

$$f ( t ) = \left\{ \begin{array} { c c } \frac { t } { 120 } & 4 \leqslant t \leqslant 16 \\ 0 & \text { otherwise } \end{array} \right.$$

1. Show that the cumulative distribution function is given by $$\mathrm { F } ( t ) = \left\{ \begin{array} { c r } 0 & t < 4 \\ \frac { t ^ { 2 } } { 240 } - c & 4 \leqslant t \leqslant 16 \\ 1 & t > 16 \end{array} \right.$$ where the value of \(c\) is to be found.
2. Find the exact probability that the amount of time Lloyd waits to be served is between 5 and 10 minutes.
3. Find the median of \(T\).
4. Find the value of \(k\) such that $$\mathrm { P } ( T < k ) = \frac { 2 } { 3 } \mathrm { P } ( T > k )$$ giving your answer to 3 significant figures.

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

$$\mathrm { F } ( x ) = \left\{ \begin{array} { c c } 0 & x < 4 \\ p x - k \sqrt { x } & 4 \leqslant x \leqslant 9 \\ 1 & x > 9 \end{array} \right.$$ where \(p\) and \(k\) are constants.

1. Find the value of \(p\) and the value of \(k\). Given that \(\mathrm { E } ( X ) = \frac { 119 } { 18 }\)
2. show that \(\operatorname { Var } ( X ) = 2.05\) to 3 significant figures.
3. Write down the mode of \(X\).
4. Find the exact value of the constant \(a\) such that \(\mathrm { P } ( X \leqslant a ) = \frac { 7 } { 27 }\)

The graph shows the probability density function \(\mathrm { f } ( x )\) of the continuous random variable \(X\) \includegraphics[max width=\textwidth, alt={}, center]{128c408d-3e08-4f74-8f19-d33ecd5c882f-04_951_1365_322_331}

1. Find \(\mathrm { P } ( X < 4 )\)
2. Specify the cumulative distribution function of \(X\) for \(7 \leqslant x \leqslant 11\)

1. A random variable \(X\) has probability density function given by

$$f ( x ) = \left\{ \begin{array} { c c } 0.8 - 6.4 x ^ { - 3 } & 2 \leqslant x \leqslant 4 \\ 0 & \text { otherwise } \end{array} \right.$$ The median of \(X\) is \(m\)

1. Show that \(m ^ { 3 } - 3.625 m ^ { 2 } + 4 = 0\)
2. 1. Find \(\mathrm { f } ^ { \prime } ( x )\)
   2. Explain why the mode of \(X\) is 4 Given that \(\mathrm { E } \left( X ^ { 2 } \right) = 10.5\) to 3 significant figures,
3. find \(\operatorname { Var } ( X )\), showing your working clearly.

1. A continuous random variable \(X\) has probability density function

$$f ( x ) = \left\{ \begin{array} { c c } \frac { x } { 16 } \left( 9 - x ^ { 2 } \right) & 1 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{array} \right.$$

1. Find the cumulative distribution function of \(X\)
2. Calculate \(\mathrm { P } ( X > 1.8 )\)
3. Use calculus to find \(\mathrm { E } \left( \frac { 3 } { X } + 2 \right)\)
4. Show that the mode of \(X\) is \(\sqrt { 3 }\)

1. A continuous random variable \(X\) has cumulative distribution function \(\mathrm { F } ( x )\) given by

$$\mathrm { F } ( x ) = \left\{ \begin{array} { c r } 0 & x < - 1 \\ \frac { 1 } { 5 } ( x + 1 ) ^ { 2 } & - 1 \leqslant x \leqslant 0 \\ 1 - \frac { 1 } { 20 } ( 4 - x ) ^ { 2 } & 0 < x \leqslant 4 \\ 1 & x > 4 \end{array} \right.$$

1. Find the probability density function, \(\mathrm { f } ( x )\)
2. 1. Sketch \(\mathrm { f } ( x )\)
   2. Hence describe the skewness of the distribution.
3. Find, to 3 significant figures, the value of \(c\) such that $$\mathrm { P } ( 1 < X < c ) = \mathrm { P } ( c < X < 2 )$$

1. The continuous random variable \(Y\) has probability density function

$$f ( y ) = \left\{ \begin{array} { c c } \frac { 1 } { 24 } ( y + 2 ) ( 4 - y ) & 0 \leqslant y \leqslant 3 \\ 0 & \text { otherwise } \end{array} \right.$$

1. Show that the mode of \(Y\) is 1 , justifying your reasoning. Given that \(\mathrm { P } ( Y < 1 ) = \frac { 13 } { 36 }\)
2. determine whether the median of \(Y\) is less than, equal to, or greater than 2 Give a reason for your answer. Given that \(\mathrm { E } \left( Y ^ { 2 } \right) = \frac { 213 } { 80 }\)
3. find, using algebraic integration, \(\operatorname { Var } ( 2 Y )\)

1. The continuous random variable \(X\) has probability density function

$$f ( x ) = \begin{cases} \frac { 1 } { 18 } ( 11 - 2 x ) & 1 \leqslant x \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$

1. Find \(\mathrm { P } ( \mathrm { X } < 3 )\)
2. State, giving a reason, whether the upper quartile of \(X\) is greater than 3, less than 3 or equal to 3 Given that \(\mathrm { E } ( \mathrm { X } ) = \frac { 9 } { 4 }\)
3. use algebraic integration to find \(\operatorname { Var } ( \mathrm { X } )\) The cumulative distribution function of \(X\) is given by $$F ( x ) = \left\{ \begin{array} { l r } 0 & x < 1 \\ \frac { 1 } { 18 } \left( 11 x - x ^ { 2 } + c \right) & 1 \leqslant x \leqslant 4 \\ 1 & x > 4 \end{array} \right.$$
4. Show that \(\mathrm { c } = - 10\)
5. Find the median of \(X\), giving your answer to 3 significant figures. \section*{Q uestion 2 continued}

The continuous random variable \(X\) is uniformly distributed over the interval \([ - 3,5 ]\).

1. Sketch the probability density function \(\mathrm { f } ( \mathrm { x } )\) of X .
2. Find the value of k such that \(\mathrm { P } ( \mathrm { X } < 2 [ \mathrm { k } - \mathrm { X } ] ) = 0.25\)
3. Use algebraic integration to show that \(\mathrm { E } \left( \mathrm { X } ^ { 3 } \right) = 17\)