5.03a Continuous random variables: pdf and cdf

1 The continuous random variable \(X\) has probability density function $$f ( x ) = \begin{cases} \frac { 1 } { 5 } & 1 \leq x \leq 6 \\ 0 & \text { otherwise } \end{cases}$$ Find \(\mathrm { P } ( X \geq 3 )\) Circle your answer. \(\frac { 1 } { 5 } \quad \frac { 2 } { 5 } \quad \frac { 3 } { 5 } \quad \frac { 4 } { 5 }\)

6 The distance, \(X\) metres, between successive breaks in a water pipe is modelled by an exponential distribution. The mean of \(X\) is 25 The distance between two successive breaks is measured. A water pipe is given a 'Red' rating if the distance is less than \(d\) metres. The government has introduced a new law changing \(d\) to 2
Before the government introduced the new law, the probability that a water pipe is given a 'Red' rating was 0.05 6

1. Explain whether or not the probability that a water pipe is given a 'Red' rating has increased as a result of the new law.
   6
2. Find the probability density function of the random variable \(X\). 6
3. After investigation, the distances between successive breaks in water pipes are found to have a standard deviation of 5 metres. Explain whether or not the use of an exponential model in parts (a) and (b) is appropriate.
   [0pt] [2 marks]

9 The continuous random variable \(X\) has the cumulative distribution function shown below. $$\mathrm { F } ( x ) = \left\{ \begin{array} { c c } 0 & x < 0 \\ \frac { 1 } { 62 } \left( 4 x ^ { 3 } + 6 x ^ { 2 } + 3 x \right) & 0 \leq x \leq 2 \\ 1 & x > 2 \end{array} \right.$$ The discrete random variable \(Y\) has the probability distribution shown below. The random variables \(X\) and \(Y\) are independent.
Find the exact value of \(\mathrm { E } \left( X ^ { 3 } + Y \right)\).

5 The continuous random variable \(X\) has cumulative distribution function $$\mathrm { F } ( x ) = \left\{ \begin{array} { c l } 0 & x \leq 1 \\ \frac { 1 } { 10 } x - \frac { 1 } { 10 } & 1 < x \leq 6 \\ \frac { 1 } { 90 } x ^ { 2 } + \frac { 1 } { 10 } & 6 < x \leq 9 \\ 1 & x > 9 \end{array} \right.$$ 5

1. Find the probability density function \(\mathrm { f } ( x )\) 5
2. Show that \(\operatorname { Var } ( X ) = \frac { 6737 } { 1200 }\) \includegraphics[max width=\textwidth, alt={}, center]{3ef4c3fd-cbf0-4ac0-a072-a07d763fd50a-07_2488_1716_219_153}

2 The random variable \(X\) has probability density function $$f ( x ) = \begin{cases} 1 & 0 < x \leq \frac { 1 } { 2 } \\ \frac { 3 } { 8 } x ^ { - 2 } & \frac { 1 } { 2 } < x \leq \frac { 3 } { 2 } \\ 0 & \text { otherwise } \end{cases}$$ Find \(\mathrm { P } ( X < 1 )\) Circle your answer.
[0pt] [1 mark] \(\frac { 1 } { 8 }\) \(\frac { 3 } { 8 }\) \(\frac { 5 } { 8 }\) \(\frac { 7 } { 8 }\) \includegraphics[max width=\textwidth, alt={}, center]{62cee897-6eac-40b3-84c1-a0d165ba6903-03_2488_1718_219_153}

3 The random variable \(X\) has an exponential distribution with probability density function \(\mathrm { f } ( x ) = \lambda \mathrm { e } ^ { - \lambda x }\) where \(x \geq 0\) 3

1. Show that the cumulative distribution function, for \(x \geq 0\), is given by \(\mathrm { F } ( x ) = 1 - \mathrm { e } ^ { - \lambda x }\) [0pt] [3 marks]
   3
2. Given that \(\lambda = 2\), find \(\mathrm { P } ( X > 1 )\), giving your answer to three decimal places.

8 The continuous random variable \(X\) has cumulative distribution function \(\mathrm { F } ( x )\) where $$\mathrm { F } ( x ) = \begin{cases} 0 & x = 0 \\ \mathrm { e } ^ { k x } - 1 & 0 \leq x \leq 5 \\ 1 & x > 5 \end{cases}$$ 8

1. Show that \(k = \frac { 1 } { 5 } \ln 2\) [0pt] [2 marks]
   8
2. Show that the median of \(X\) is \(a \frac { \ln b } { \ln 2 } - c\), where \(a , b\) and \(c\) are integers to be found.
   

   8	Show that the mean of \(X\) is \(p - \frac { q } { \ln 2 }\), where \(p\) and \(q\) are integers to be found.

9 Lianne models the maximum time in hours that a rechargeable battery can be used, before needing to be recharged, with a rectangular distribution with values between 8 and 12 9

1. The probability that the maximum time the battery can be used before needing to be recharged is more than 10.5 hours is equal to \(p\) Lianne will only buy the battery if \(p\) is more than 0.4
   Determine whether Lianne will buy the battery.
   [0pt] [2 marks]
   9
2. A histogram is plotted for 100 recharges showing the maximum time the battery can be used before needing to be recharged. \includegraphics[max width=\textwidth, alt={}, center]{62cee897-6eac-40b3-84c1-a0d165ba6903-15_670_1186_404_427} Explain why the model used in part (a) may not be valid and suggest the name of a different distribution that could be used to model the maximum time between recharges. \includegraphics[max width=\textwidth, alt={}, center]{62cee897-6eac-40b3-84c1-a0d165ba6903-16_2488_1732_219_139}
   \includegraphics[max width=\textwidth, alt={}]{62cee897-6eac-40b3-84c1-a0d165ba6903-20_2496_1721_214_148}

6 A game consists of two rounds. The first round of the game uses a random number generator to output the score \(X\), a real number between 0 and 10 6

1. Find \(\mathrm { P } ( X > 4 )\) 6
2. The second round of the game uses an unbiased dice, with faces numbered 1 to 6 , to give the score \(Y\) The variables \(X\) and \(Y\) are independent.
   6 (b) (i) Find the mean total score of the game.
   6 (b) (ii) Find the variance of the total score of the game.

8 The continuous random variable \(X\) has probability density function $$f ( x ) = \begin{cases} k \sin 2 x & 0 \leq x \leq \frac { \pi } { 6 } \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant. 8

1. Show that \(k = 4\) 8
2. Find the cumulative distribution function \(\mathrm { F } ( x )\) 8
3. Find the median of \(X\), giving your answer to three significant figures. 8
4. Find the mean of \(X\) giving your answer in the form \(\frac { 1 } { a } ( b \sqrt { 3 } - \pi )\) where \(a\) and \(b\) are integers. \includegraphics[max width=\textwidth, alt={}, center]{1e2fdd33-afa4-486f-a9e2-1d425ed14eee-14_2492_1721_217_150}

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256 2 The random variable \(T\) has an exponential distribution with mean 2 Find \(\mathrm { P } ( T \leq 1.4 )\) Circle your answer. \(\mathrm { e } ^ { - 2.8 }\) \(\mathrm { e } ^ { - 0.7 }\) \(1 - e ^ { - 0.7 }\) \(1 - \mathrm { e } ^ { - 2.8 }\) The continuous random variable \(Y\) has cumulative distribution function $$\mathrm { F } ( y ) = \left\{ \begin{array} { l r } 0 & y < 2 \\ - \frac { 1 } { 9 } y ^ { 2 } + \frac { 10 } { 9 } y - \frac { 16 } { 9 } & 2 \leq y < 5 \\ 1 & y \geq 5 \end{array} \right.$$ Find the median of \(Y\) Circle your answer. 2 \(\frac { 10 - 3 \sqrt { 2 } } { 2 }\) \(\frac { 7 } { 2 }\) \(\frac { 10 + 3 \sqrt { 2 } } { 2 }\) Turn over for the next question 4 Research has shown that the mean number of volcanic eruptions on Earth each day is 20 Sandra records 162 volcanic eruptions during a period of one week. Sandra claims that there has been an increase in the mean number of volcanic eruptions per week. Test Sandra's claim at the \(5 \%\) level of significance.
5 The continuous random variable \(X\) has probability density function $$f ( x ) = \begin{cases} \frac { 1 } { 6 } e ^ { \frac { x } { 3 } } & 0 \leq x \leq \ln 27 \\ 0 & \text { otherwise } \end{cases}$$ Show that the mean of \(X\) is \(\frac { 3 } { 2 } ( \ln 27 - 2 )\) 6 Over time it has been accepted that the mean retirement age for professional baseball players is 29.5 years old. Imran claims that the mean retirement age is no longer 29.5 years old.
He takes a random sample of 5 recently retired professional baseball players and records their retirement ages, \(x\). The results are $$\sum x = 152.1 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 7.81$$ 6

1. State an assumption that you should make about the distribution of the retirement ages to investigate Imran's claim. 6
2. Investigate Imran's claim, using the 10\% level of significance.

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256 2 The random variable \(T\) has an exponential distribution with mean 2 Find \(\mathrm { P } ( T \leq 1.4 )\) Circle your answer. \(\mathrm { e } ^ { - 2.8 }\) \(\mathrm { e } ^ { - 0.7 }\) \(1 - e ^ { - 0.7 }\) \(1 - \mathrm { e } ^ { - 2.8 }\) The continuous random variable \(Y\) has cumulative distribution function $$\mathrm { F } ( y ) = \left\{ \begin{array} { l r } 0 & y < 2 \\ - \frac { 1 } { 9 } y ^ { 2 } + \frac { 10 } { 9 } y - \frac { 16 } { 9 } & 2 \leq y < 5 \\ 1 & y \geq 5 \end{array} \right.$$ Find the median of \(Y\) Circle your answer. 2 \(\frac { 10 - 3 \sqrt { 2 } } { 2 }\) \(\frac { 7 } { 2 }\) \(\frac { 10 + 3 \sqrt { 2 } } { 2 }\) Turn over for the next question 4 Research has shown that the mean number of volcanic eruptions on Earth each day is 20 Sandra records 162 volcanic eruptions during a period of one week. Sandra claims that there has been an increase in the mean number of volcanic eruptions per week. Test Sandra's claim at the \(5 \%\) level of significance.
5 The continuous random variable \(X\) has probability density function $$f ( x ) = \begin{cases} \frac { 1 } { 6 } e ^ { \frac { x } { 3 } } & 0 \leq x \leq \ln 27 \\ 0 & \text { otherwise } \end{cases}$$ Show that the mean of \(X\) is \(\frac { 3 } { 2 } ( \ln 27 - 2 )\) 6 Over time it has been accepted that the mean retirement age for professional baseball players is 29.5 years old. Imran claims that the mean retirement age is no longer 29.5 years old.
He takes a random sample of 5 recently retired professional baseball players and records their retirement ages, \(x\). The results are $$\sum x = 152.1 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 7.81$$ 6

1. State an assumption that you should make about the distribution of the retirement ages to investigate Imran's claim. 6
2. Investigate Imran's claim, using the 10\% level of significance.

2 The continuous random variable \(X\) takes values in the interval \(- 1 \leq x \leq 1\) and has probability density function $$f ( x ) = \left\{ \begin{array} { l r } a & - 1 \leq x < 0 \\ a + x ^ { 2 } & 0 \leq x \leq 1 \end{array} \right.$$ where \(a\) is a constant.

1. (A) Sketch the probability density function.
   (B) Show that \(a = \frac { 1 } { 3 }\).
2. Find
   (A) \(\mathrm { P } \left( X < \frac { 1 } { 2 } \right)\),
   (B) the mean of \(X\).
3. Show that the median of \(X\) satisfies the equation \(2 m ^ { 3 } + 2 m - 1 = 0\).

3. A string of length 60 cm is cut a random point.

1. Name a distribution, including parameters, that can be used to model the length of the longer piece of string and find its mean and variance.
2. The longer string is shaped to form the perimeter of a circle. Find the probability that the area of the circle is greater than \(100 \mathrm {~cm} ^ { 2 }\).

4 The continuous random variable \(X\) has probability density function $$\mathrm { f } ( x ) = \begin{cases} \frac { k } { x ^ { n } } & x \geqslant 1 \\ 0 & \text { otherwise } \end{cases}$$ where \(n\) and \(k\) are constants and \(n\) is an integer greater than 1 .

1. Find \(k\) in terms of \(n\).
2. 1. When \(n = 4\), find the cumulative distribution function of \(X\).
   2. Hence determine \(\mathrm { P } ( X > 7 \mid X > 5 )\) when \(n = 4\).
3. Determine the values of \(n\) for which \(\operatorname { Var } ( X )\) is not defined.

1. A biased die with six faces is rolled. The discrete random variable \(X\) represents the score which is uppermost. The cumulative distribution function of \(X\) is shown in the table below.

1. Find the value of the constant \(k\)
2. Find the probability distribution of \(X\) A biased die with eight faces is rolled. The discrete random variable \(Y\) represents the score which is uppermost. The probability distribution of \(Y\) is shown in the table below, where \(a\) and \(b\) are constants.

   \(y\)	1	2	3	4	5	6	7	8
   \(\mathrm { P } ( Y = y )\)	\(a\)	\(a\)	\(a\)	\(b\)	\(b\)	\(b\)	0.11	0.05

   Given that \(\mathrm { E } ( Y ) = 4.02\)
3. form and solve two equations in \(a\) and \(b\) to show that \(a = 0.15\) You must show your working.
   (Solutions relying on calculator technology are not acceptable.)
4. Show that \(\mathrm { E } \left( Y ^ { 2 } \right) = 20.7\)
5. Find \(\operatorname { Var } ( 5 - 2 Y )\) These dice are each rolled once. The scores on the two dice are independent.
6. Find the probability that the sum of these two scores is 3

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

$$f ( x ) = \begin{cases} \frac { 1 } { 4 } ( 3 - x ) & 1 \leqslant x \leqslant 2 \\ \frac { 1 } { 4 } & 2 < x \leqslant 3 \\ \frac { 1 } { 4 } ( x - 2 ) & 3 < x \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$ The cumulative distribution function of \(X\) is \(\mathrm { F } ( x )\)

1. Show that \(\mathrm { F } ( x ) = \frac { 1 } { 4 } \left( 3 x - \frac { x ^ { 2 } } { 2 } \right) - \frac { 5 } { 8 }\) for \(1 \leqslant x \leqslant 2\)
2. Find \(\mathrm { F } ( x )\) for all values of \(x\)
3. Find \(\mathrm { P } ( 1.2 < X < 3.1 )\)

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

$$f ( x ) = \begin{cases} a x & 0 \leqslant x \leqslant 4 \\ b x + c & 4 < x \leqslant 8 \\ 0 & \text { otherwise } \end{cases}$$ where \(a\), \(b\) and \(c\) are constants. \begin{figure}[h] \end{figure} Figure 1 shows the graph of the probability density function \(\mathrm { f } ( x )\) The graph consists of two straight line segments of equal length joined at the point where \(x = 4\)

1. Show that \(a = \frac { 1 } { 16 }\)
2. Hence find
   1. the value of \(b\)
   2. the value of \(c\)
3. Using algebraic integration, show that \(\operatorname { Var } ( X ) = \frac { 8 } { 3 }\)
4. Find, to 2 decimal places, the lower quartile and the upper quartile of \(X\) A statistician claims that $$\mathrm { P } ( - \sigma < X - \mu < \sigma ) > 0.5$$ where \(\mu\) and \(\sigma\) are the mean and standard deviation of \(X\)
5. Show that the statistician's claim is correct.

7 The number of goals scored by a hockey team in an interval of time of length \(t\) minutes follows a Poisson distribution with mean \(\frac { 1 } { 24 } t\). The random variable \(T\) is defined as the length of time, in minutes, between successive goals.

1. (a) Show that \(\mathrm { P } ( T < t ) = 1 - \mathrm { e } ^ { - \frac { 1 } { 24 } t }\) for \(t \geqslant 0\).
   (b) Hence find the probability density function of \(T\).
2. Find the exact value of the interquartile range of \(T\).

1 The random variable \(X\) has probability density function \(\mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = \begin{cases} k \mathrm { e } ^ { - k x } & x \geqslant 0 , \\ 0 & x < 0 , \end{cases}$$ and \(k\) is a positive constant.

1. Show that the moment generating function of \(X\) is \(\mathrm { M } _ { X } ( t ) = k ( k - t ) ^ { - 1 } , t < k\).
2. Use the moment generating function to find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).

6 The lengths of time, in years, that sales representatives for a certain company keep their company cars may be modelled by the distribution with probability density function \(\mathrm { f } ( x )\), where $$f ( x ) = \begin{cases} \frac { 4 } { 27 } x ^ { 2 } ( 3 - x ) & 0 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{cases}$$

1. Draw a sketch of this probability density function.
2. Calculate the mean and the mode of \(X\).
3. Comment briefly on the values obtained in part (ii) in relation to the sketch in part (i).
4. Given that \(\sigma ^ { 2 } = 0.36\), find \(\mathrm { P } ( | X - \mu | < \sigma )\), where \(\mu\) and \(\sigma ^ { 2 }\) denote the mean and the variance of \(X\) respectively.

5 The discrete random variable \(X\) has probability generating function given by $$\mathrm { G } _ { X } ( t ) = k \left( 5 t ^ { - 1 } + 3 + 2 t ^ { 2 } \right) ,$$ where \(k\) is a constant.

1. Find
   1. the value of \(k\),
   2. the modal value of \(X\).
   3. The random variables \(X _ { 1 }\) and \(X _ { 2 }\) are independent observations of \(X\).
      (a) Write down the probability generating function of \(Y\), where \(Y = X _ { 1 } + X _ { 2 }\).
      (b) Use your answer to part (ii)(a) to find \(\mathrm { E } ( Y )\) and \(\operatorname { Var } ( Y )\).

6 A rectangle of area \(Y \mathrm {~m} ^ { 2 }\) has a perimeter of 16 m and a side of length \(X \mathrm {~m}\), where \(X\) is a random variable with probability density function, f, given by $$f ( x ) = \begin{cases} \frac { 1 } { 2 } & 0 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{cases}$$

1. Obtain the cumulative distribution function, F , of \(X\).
2. Show that $$16 - Y = ( 4 - X ) ^ { 2 }$$ and deduce that the probability density function of the random variable \(Y\) is $$g ( y ) = \begin{cases} \frac { 1 } { 4 \sqrt { 16 - y } } & 0 \leqslant y \leqslant 12 \\ 0 & \text { otherwise } \end{cases}$$
3. Find the median of \(Y\).
4. Find \(\mathrm { E } ( Y )\).