Single-piece PDF with k

6 A random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} \frac { k } { x ^ { 2 } } & 1 \leqslant x \leqslant a
0 & \text { otherwise } \end{cases}$$ where \(k\) and \(a\) are positive constants.

1. Show that \(k = \frac { a } { a - 1 }\).
2. Find \(\mathrm { E } ( X )\) in terms of \(a\).
3. Find the 60th percentile of \(X\) in terms of \(a\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

7 The probability density function, f , of a random variable \(X\) is given by $$f ( x ) = \begin{cases} k ( 1 + \cos x ) & 0 \leqslant x \leqslant \pi
0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = \frac { 1 } { \pi }\).
2. Verify that the median of \(X\) lies between 0.83 and 0.84 .
3. Find the exact value of \(\mathrm { E } ( X )\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

5 A random variable \(X\) has probability density function f given by $$f ( x ) = \begin{cases} a x - x ^ { 3 } & 0 \leqslant x \leqslant \sqrt { 2 }
0 & \text { otherwise } \end{cases}$$ where \(a\) is a constant.

1. Show that \(a = 2\) .
2. Find the median of \(X\) .
3. Find the exact value of \(\mathrm { E } ( X )\).

6 The time, \(T\) hours, spent by people on a visit to a museum has probability density function $$\mathrm { f } ( t ) = \begin{cases} k t \left( 16 - t ^ { 2 } \right) & 0 \leqslant t \leqslant 4
0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = \frac { 1 } { 64 }\).
2. Calculate the probability that two randomly chosen people each spend less than 1 hour on a visit to the museum.
3. Find the mean time spent on a visit to the museum.

7 The probability density function of the random variable \(X\) is given by $$f ( x ) = \begin{cases} \frac { 3 } { 4 } x ( c - x ) & 0 \leqslant x \leqslant c
0 & \text { otherwise } \end{cases}$$ where \(c\) is a constant.

1. Show that \(c = 2\).
2. Sketch the graph of \(y = \mathrm { f } ( x )\) and state the median of \(X\).
3. Find \(\mathrm { P } ( X < 1.5 )\).
4. Hence write down the value of \(\mathrm { P } ( 0.5 < X < 1 )\).

6 In each turn of a game, a coin is pushed and slides across a table. The distance, \(X\) metres, travelled by the coin has probability density function given by $$f ( x ) = \begin{cases} k x ^ { 2 } ( 2 - x ) & 0 \leqslant x \leqslant 2
0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.

1. State the greatest possible distance travelled by the coin in one turn.
2. Show that \(k = \frac { 3 } { 4 }\).
3. Find the mean distance travelled by the coin in one turn.
4. Out of 400 turns, find the expected number of turns in which the distance travelled by the coin is less than 1 metre.

6
\includegraphics[max width=\textwidth, alt={}, center]{395f7f2c-42db-4fb6-9b22-3b0f46ad16d3-08_355_670_260_735} The diagram shows the graph of the probability density function, f , of a continuous random variable \(X\), where f is defined by $$\mathrm { f } ( x ) = \begin{cases} k \left( x - x ^ { 2 } \right) & 0 \leqslant x \leqslant 1
0 & \text { otherwise } \end{cases}$$

1. Show that the value of the constant \(k\) is 6 .
2. State the value of \(\mathrm { E } ( X )\) and find \(\operatorname { Var } ( X )\).
3. Find \(\mathrm { P } ( 0.4 < X < 2 )\).

7 A random variable \(X\) has probability density function defined by $$f ( x ) = \begin{cases} k \left( \frac { 1 } { x ^ { 2 } } + \frac { 1 } { x ^ { 3 } } \right) & 1 \leqslant x \leqslant 2
0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = \frac { 8 } { 7 }\).
2. Find \(\mathrm { E } ( X )\).
3. Three values of \(X\) are chosen at random. Find the probability that one of these values is less than 1.5 and the other two are greater than 1.5.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

7 A random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} k ( 1 - x ) & - 1 \leqslant x \leqslant 1
0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = \frac { 1 } { 2 }\).
2. Find \(\mathrm { P } \left( X > \frac { 1 } { 2 } \right)\).
3. Find the mean of \(X\).
4. Find \(a\) such that \(\mathrm { P } ( X < a ) = \frac { 1 } { 4 }\).

4 The random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} \frac { k } { ( x + 1 ) ^ { 2 } } & 0 \leqslant x \leqslant 1
0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = 2\).
2. Find \(a\) such that \(\mathrm { P } ( X < a ) = \frac { 1 } { 5 }\).
3. 
   \includegraphics[max width=\textwidth, alt={}, center]{18cef198-5ca2-4700-88e9-1a2bd55f841e-2_367_524_1548_849} The diagram shows the graph of \(y = \mathrm { f } ( x )\). The median of \(X\) is denoted by \(m\). Use the diagram to explain whether \(m < 0.5\), \(m = 0.5\) or \(m > 0.5\).

7

1. The probability density function of the random variable \(X\) is given by $$f ( x ) = \begin{cases} k x ( 4 - x ) & 0 \leqslant x \leqslant 2
   0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.
   1. Show that \(k = \frac { 3 } { 16 }\).
   2. Find \(\mathrm { E } ( X )\).
2. The random variable \(Y\) has the following properties.
   • \(Y\) takes values between 0 and 5 only.
   • The probability density function of \(Y\) is symmetrical.
   Given that \(\mathrm { P } ( Y < a ) = 0.2\), find \(\mathrm { P } ( 2.5 < Y < 5 - a )\) illustrating your method with a sketch on the axes provided.
   \includegraphics[max width=\textwidth, alt={}, center]{cea87af9-4b2a-4297-91e9-4eb5744b9e48-11_369_837_621_694}
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

7 The queuing time, \(T\) minutes, for a person queuing at a supermarket checkout has probability density function given by $$f ( t ) = \begin{cases} c t \left( 25 - t ^ { 2 } \right) & 0 \leqslant t \leqslant 5
0 & \text { otherwise } \end{cases}$$ where \(c\) is a constant.

1. Show that the value of \(c\) is \(\frac { 4 } { 625 }\).
2. Find the probability that a person will have to queue for between 2 and 4 minutes.
3. Find the mean queuing time.

7 The random variable \(X\) denotes the number of hours of cloud cover per day at a weather forecasting centre. The probability density function of \(X\) is given by $$f ( x ) = \begin{cases} \frac { ( x - 18 ) ^ { 2 } } { k } & 0 \leqslant x \leqslant 24
0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = 2016\).
2. On how many days in a year of 365 days can the centre expect to have less than 2 hours of cloud cover?
3. Find the mean number of hours of cloud cover per day.

5 The random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} 4 x ^ { k } & 0 \leqslant x \leqslant 1
0 & \text { otherwise } \end{cases}$$ where \(k\) is a positive constant.

1. Show that \(k = 3\).
2. Show that the mean of \(X\) is 0.8 and find the variance of \(X\).
3. Find the upper quartile of \(X\).
4. Find the interquartile range of \(X\).

7 If Usha is stung by a bee she always develops an allergic reaction. The time taken in minutes for Usha to develop the reaction can be modelled using the probability density function given by $$\mathrm { f } ( t ) = \begin{cases} \frac { k } { t + 1 } & 0 \leqslant t \leqslant 4
0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = \frac { 1 } { \ln 5 }\).
2. Find the probability that it takes more than 3 minutes for Usha to develop a reaction.
3. Find the median time for Usha to develop a reaction.

5 The time in minutes taken by candidates to answer a question in an examination has probability density function given by $$\mathrm { f } ( t ) = \begin{cases} k \left( 6 t - t ^ { 2 } \right) & 3 \leqslant t \leqslant 6
0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = \frac { 1 } { 18 }\).
2. Find the mean time.
3. Find the probability that a candidate, chosen at random, takes longer than 5 minutes to answer the question.
4. Is the upper quartile of the times greater than 5 minutes, equal to 5 minutes or less than 5 minutes? Give a reason for your answer.

5 The random variable \(T\) denotes the time in seconds for which a firework burns before exploding. The probability density function of \(T\) is given by $$\mathrm { f } ( t ) = \begin{cases} k \mathrm { e } ^ { 0.2 t } & 0 \leqslant t \leqslant 5
0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = \frac { 1 } { 5 ( \mathrm { e } - 1 ) }\).
2. Sketch the probability density function.
3. \(80 \%\) of fireworks burn for longer than a certain time before they explode. Find this time.

5 The time, in minutes, taken by volunteers to complete a task is modelled by the random variable \(X\) with probability density function given by $$\mathrm { f } ( x ) = \begin{cases} \frac { k } { x ^ { 4 } } & x \geqslant 1
0 & \text { otherwise. } \end{cases}$$

1. Show that \(k = 3\).
2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).

6 At a certain shop the weekly demand, in kilograms, for flour is modelled by the random variable \(X\) with probability density function given by $$f ( x ) = \begin{cases} k x ^ { - \frac { 1 } { 2 } } & 4 \leqslant x \leqslant 25
0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = \frac { 1 } { 6 }\).
2. Calculate the mean weekly demand for flour at the shop.
3. At the beginning of one week, the shop has 20 kg of flour in stock. Find the probability that this will not be enough to meet the demand for that week.
4. Give a reason why the model may not be realistic.

7
\includegraphics[max width=\textwidth, alt={}, center]{7333c047-edad-4385-b3f8-248e8725cfcb-3_412_718_1037_715} A random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} k \sin x & 0 \leqslant x \leqslant \frac { 2 } { 3 } \pi
0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant, as shown in the diagram.

1. Show that \(k = \frac { 2 } { 3 }\).
2. Show that the median of \(X\) is 1.32 , correct to 3 significant figures.
3. Find \(\mathrm { E } ( X )\).

5 A random variable \(X\) has probability density function given by $$\mathrm { f } ( x ) = \begin{cases} \frac { k } { x ^ { 3 } } & x \geqslant 1
0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = 2\).
2. Find \(\mathrm { P } ( 1 \leqslant X \leqslant 2 )\).
3. Find \(\mathrm { E } ( X )\).

7 A random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} \frac { k } { x } & 1 \leqslant x \leqslant a
0 & \text { otherwise } \end{cases}$$ where \(k\) and \(a\) are positive constants.

1. Show that \(k = \frac { 1 } { \ln a }\).
2. Find \(\mathrm { E } ( X )\) in terms of \(a\).
3. Find the median of \(X\) in terms of \(a\).

6 The waiting time, \(T\) minutes, for patients at a doctor's surgery has probability density function given by $$\mathrm { f } ( t ) = \begin{cases} k \left( 225 - t ^ { 2 } \right) & 0 \leqslant t \leqslant 15
0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = \frac { 1 } { 2250 }\).
2. Find the probability that a patient has to wait for more than 10 minutes.
3. Find the mean waiting time.

5 The time, \(T\) minutes, taken by people to complete a test has probability density function given by $$\mathrm { f } ( t ) = \begin{cases} k \left( 10 t - t ^ { 2 } \right) & 5 \leqslant t \leqslant 10
0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = \frac { 3 } { 250 }\).
2. Find \(\mathrm { E } ( T )\).
3. Find the probability that a randomly chosen value of \(T\) lies between \(\mathrm { E } ( T )\) and the median of \(T\). [3]
4. State the greatest possible length of time taken to complete the test.
   \(6 X\) and \(Y\) are independent random variables with distributions \(\operatorname { Po } ( 1.6 )\) and \(\operatorname { Po } ( 2.3 )\) respectively.

6 The time, in minutes, taken by people to complete a test is modelled by the continuous random variable \(X\) with probability density function given by $$f ( x ) = \begin{cases} \frac { k } { x ^ { 2 } } & 5 \leqslant x \leqslant 10
0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = 10\).
2. Show that \(\mathrm { E } ( X ) = 10 \ln 2\).
3. Find \(\mathrm { P } ( X > 9 )\).
4. Given that \(\mathrm { P } ( X < a ) = 0.6\), find \(a\).