5.03b Solve problems: using pdf

3 \includegraphics[max width=\textwidth, alt={}, center]{823cc2e5-e408-4b81-ac4d-7e9f584107cc-06_558_1077_260_523} The diagram shows the graph of the probability density function, f, of a random variable \(X\) that takes values between \(x = 0\) and \(x = 3\) only. The graph is symmetrical about the line \(x = 1.5\).

1. It is given that \(\mathrm { P } ( X < 0.6 ) = a\) and \(\mathrm { P } ( 0.6 < X < 1.2 ) = b\). Find \(\mathrm { P } ( 0.6 < X < 1.8 )\) in terms of \(a\) and \(b\).
2. It is now given that the equation of the probability density function of \(X\) is $$f ( x ) = \begin{cases} k x ^ { 2 } ( 3 - x ) ^ { 2 } & 0 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.
   1. Show that \(k = \frac { 10 } { 81 }\).
   2. Find \(\operatorname { Var } ( X )\).

6 The length of time, \(T\) minutes, that a passenger has to wait for a bus at a certain bus stop is modelled by the probability density function given by $$\mathrm { f } ( t ) = \begin{cases} \frac { 3 } { 4000 } \left( 20 t - t ^ { 2 } \right) & 0 \leqslant t \leqslant 20 \\ 0 & \text { otherwise } \end{cases}$$

1. Sketch the graph of \(y = \mathrm { f } ( t )\).
2. Hence explain, without calculation, why \(\mathrm { E } ( T ) = 10\).
3. Find \(\operatorname { Var } ( T )\).
4. It is given that \(\mathrm { P } ( T < 10 + a ) = p\), where \(0 < a < 10\). Find \(\mathrm { P } ( 10 - a < T < 10 + a )\) in terms of \(p\).
5. Find \(\mathrm { P } ( 8 < T < 12 )\).
6. Give one reason why this model may be unrealistic.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

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 A factory is supplied with grain at the beginning of each week. The weekly demand, \(X\) thousand tonnes, for grain from this factory is a continuous random variable having the probability density function given by $$f ( x ) = \begin{cases} 2 ( 1 - x ) & 0 \leqslant x \leqslant 1 \\ 0 & \text { otherwise } \end{cases}$$ Find

1. the mean value of \(X\),
2. the variance of \(X\),
3. the quantity of grain in tonnes that the factory should have in stock at the beginning of a week, in order to be \(98 \%\) certain that the demand in that week will be met.

4 A random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} 1 - \frac { 1 } { 2 } x & 0 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{cases}$$

1. Find \(\mathrm { P } ( X > 1.5 )\).
2. Find the mean of \(X\).
3. Find the median of \(X\).

6 The length, \(X\) centimetres, of worms of a certain type is modelled by the probability density function $$f ( x ) = \begin{cases} \frac { 6 } { 125 } ( 10 - x ) ( x - 5 ) & 5 \leqslant x \leqslant 10 \\ 0 & \text { otherwise } \end{cases}$$

1. State the value of \(\mathrm { E } ( X )\).
2. Find \(\operatorname { Var } ( X )\).
3. Two worms of this type are chosen at random. Find the probability that exactly one of them has length less than 6 cm .

3 \includegraphics[max width=\textwidth, alt={}, center]{9fca48da-82c3-4ce1-9e0c-93eb9a920f9d-05_456_668_260_735} The random variable \(X\) takes values in the range \(1 \leqslant x \leqslant p\), where \(p\) is a constant. The graph of the probability density function of \(X\) is shown in the diagram.

1. Show that \(p = 2\).
2. Find \(\mathrm { E } ( X )\).

6 Alethia models the length of time, in minutes, by which her train is late on any day by the random variable \(X\) with probability density function given by $$f ( x ) = \begin{cases} \frac { 3 } { 8000 } ( x - 20 ) ^ { 2 } & 0 \leqslant x \leqslant 20 \\ 0 & \text { otherwise } \end{cases}$$

1. Find the probability that the train is more than 10 minutes late on each of two randomly chosen days.
2. Find \(\mathrm { E } ( X )\).
3. The median of \(X\) is denoted by \(m\). Show that \(m\) satisfies the equation \(( m - 20 ) ^ { 3 } = - 4000\), and hence find \(m\) correct to 3 significant figures.
4. State one way in which Alethia's model may be unrealistic.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

6 A random variable \(X\) has probability density function f . The graph of \(\mathrm { f } ( x )\) is a straight line segment parallel to the \(x\)-axis from \(x = 0\) to \(x = a\), where \(a\) is a positive constant.

1. State, in terms of \(a\), the median of \(X\).
2. Find \(\mathrm { P } \left( X > \frac { 3 } { 4 } a \right)\).
3. Show that \(\operatorname { Var } ( X ) = \frac { 1 } { 12 } a ^ { 2 }\).
4. Given that \(\mathrm { P } ( X < b ) = p\), where \(0 < b < \frac { 1 } { 2 } a\), find \(\mathrm { P } \left( \frac { 1 } { 3 } b < X < a - \frac { 1 } { 3 } b \right)\) in terms of \(p\).

5 A random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} \frac { 3 } { 16 } \left( 4 x - x ^ { 2 } \right) & 2 \leqslant x \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$

1. Show that \(\mathrm { E } ( X ) = \frac { 11 } { 4 }\).
2. Find \(\operatorname { Var } ( X )\).
3. Given that the median of \(X\) is \(m\), find \(\mathrm { P } ( m < X < 3 )\).

7 The random variables \(X\) and \(W\) have probability density functions f and g defined as follows: $$\begin{gathered} \mathrm { f } ( x ) = \begin{cases} p \left( a ^ { 2 } - x ^ { 2 } \right) & 0 \leqslant x \leqslant a \\ 0 & \text { otherwise } \end{cases} \\ \mathrm { g } ( w ) = \begin{cases} q \left( a ^ { 2 } - w ^ { 2 } \right) & - a \leqslant w \leqslant a \\ 0 & \text { otherwise } \end{cases} \end{gathered}$$ where \(a , p\) and \(q\) are constants.

1. 1. Write down the value of \(\mathrm { P } ( X \geqslant 0 )\).
   2. Write down the value of \(\mathrm { P } ( W \geqslant 0 )\).
   3. Write down an expression for \(q\) in terms of \(p\) only.
2. Given that \(\mathrm { E } ( X ) = 3\), find the value 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.

2

1. \includegraphics[max width=\textwidth, alt={}, center]{b0960fa7-ddbe-47b7-929e-f62f72f9dc93-04_324_574_264_813} The graph of the function f is a straight line segment from \(( 0,0 )\) to \(( 2,1 )\).
   Show that \(f\) could be a probability density function.
2. \includegraphics[max width=\textwidth, alt={}, center]{b0960fa7-ddbe-47b7-929e-f62f72f9dc93-04_364_592_1466_804} The graph of the function g is a semicircle, centre \(( 0,0 )\), entirely above the \(x\)-axis.
   Given that g is a probability density function, find the radius of the semicircle.
3. \includegraphics[max width=\textwidth, alt={}, center]{b0960fa7-ddbe-47b7-929e-f62f72f9dc93-05_369_826_264_689} The time, \(X\) minutes, taken by a large number of students to complete a test has probability density function h , as shown in the diagram.
   1. Without calculation, use the diagram to explain how you can tell that the median time is less than 15 minutes.
      It is now given that $$h ( x ) = \begin{cases} \frac { 40 } { x ^ { 2 } } - \frac { 1 } { 10 } & 10 \leqslant x \leqslant 20 \\ 0 & \text { otherwise. } \end{cases}$$
   2. Find the mean time.

7

1. \includegraphics[max width=\textwidth, alt={}, center]{593c1ece-82a2-4dcd-8041-f39c98adf631-12_357_738_267_731} The diagram shows the graph of the probability density function, f , of a random variable \(X\) which takes values between 0 and 4 only. Between these two values the graph is a straight line.
   1. Show that \(\mathrm { f } ( x ) = k x\) for \(0 \leqslant x \leqslant 4\), where \(k\) is a constant to be determined.
   2. Hence, or otherwise, find \(\mathrm { E } ( X )\).
2. \includegraphics[max width=\textwidth, alt={}, center]{593c1ece-82a2-4dcd-8041-f39c98adf631-13_383_752_269_731} The diagram shows the graph of the probability density function, g , of a random variable \(W\) which takes values between 0 and \(a\) only, where \(a > 0\). Between these two values the graph is a straight line. Given that the median of \(W\) is 1 , find the value 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 \includegraphics[max width=\textwidth, alt={}, center]{10cf346f-dee2-4223-8caa-2a49f1eaa99f-10_547_880_260_621} A random variable \(X\) has probability density function f , where the graph of \(y = \mathrm { f } ( x )\) is a semicircle with centre \(( 0,0 )\) and radius \(\sqrt { \frac { 2 } { \pi } }\), entirely above the \(x\)-axis. Elsewhere \(\mathrm { f } ( x ) = 0\) (see diagram).

1. Verify that f can be a probability density function. \(A\) and \(B\) are the points where the line \(x = \sqrt { \frac { 1 } { \pi } }\) meets the \(x\)-axis and the semicircle respectively.
2. Show that angle \(A O B\) is \(\frac { 1 } { 4 } \pi\) radians and hence find \(\mathrm { P } \left( X > \sqrt { \frac { 1 } { \pi } } \right)\).

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 \(\mathrm { E } ( X )\).

5 Bottles of Lanta contain approximately 300 ml of juice. The volume of juice, in millilitres, in a bottle is \(300 + X\), where \(X\) is a random variable with probability density function given by $$f ( x ) = \begin{cases} \frac { 3 } { 4000 } \left( 100 - x ^ { 2 } \right) & - 10 \leqslant x \leqslant 10 \\ 0 & \text { otherwise } \end{cases}$$

1. Find the probability that a randomly chosen bottle of Lanta contains more than 305 ml of juice.
2. Given that \(25 \%\) of bottles of Lanta contain more than \(( 300 + p ) \mathrm { ml }\) of juice, show that $$p ^ { 3 } - 300 p + 1000 = 0 .$$
3. Given that \(p = 3.47\), and that \(50 \%\) of bottles of Lanta contain between ( \(300 - q\) ) and ( \(300 + q\) ) ml of juice, find \(q\). Justify your answer.

2 \includegraphics[max width=\textwidth, alt={}, center]{image-not-found} The diagram shows the graph of the probability density function, f , of a random variable \(X\).

1. Find the value of the constant \(k\).
2. Using this value of \(k\), find \(\mathrm { f } ( x )\) for \(0 \leqslant x \leqslant k\) and hence find \(\mathrm { E } ( X )\).
3. Find the value of \(p\) such that \(\mathrm { P } ( p < X < 1 ) = 0.25\).

6 In a game a ball is rolled down a slope and along a track until it stops. The distance, in metres, travelled by the ball is modelled by the random variable \(X\) with probability density function $$f ( x ) = \begin{cases} - k ( x - 1 ) ( x - 3 ) & 1 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.

1. Without calculation, explain why \(\mathrm { E } ( X ) = 2\).
2. Show that \(k = \frac { 3 } { 4 }\).
3. Find \(\operatorname { Var } ( X )\).
   One turn consists of rolling the ball 3 times and noting the largest value of \(X\) obtained. If this largest value is greater than 2.5, the player scores a point.
4. Find the probability that on a particular turn the player scores a point.

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.

7

1. \includegraphics[max width=\textwidth, alt={}, center]{1060d9f5-cf40-419e-b212-7266885c6617-3_465_1127_954_550} The diagram shows the graph of the probability density function of a variable \(X\). Given that the graph is symmetrical about the line \(x = 1\) and that \(\mathrm { P } ( 0 < X < 2 ) = 0.6\), find \(\mathrm { P } ( X > 0 )\).
2. A flower seller wishes to model the length of time that tulips last when placed in a jug of water. She proposes a model using the random variable \(X\) (in hundreds of hours) with probability density function given by $$f ( x ) = \begin{cases} k \left( 2.25 - x ^ { 2 } \right) & 0 \leqslant x \leqslant 1.5 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.
   1. Show that \(k = \frac { 4 } { 9 }\).
   2. Use this model to find the mean number of hours that a tulip lasts in a jug of water. The flower seller wishes to create a similar model for daffodils. She places a large number of daffodils in jugs of water and the longest time that any daffodil lasts is found to be 290 hours.
   3. Give a reason why \(\mathrm { f } ( x )\) would not be a suitable model for daffodils.
   4. The flower seller considers a model for daffodils of the form $$g ( x ) = \begin{cases} c \left( a ^ { 2 } - x ^ { 2 } \right) & 0 \leqslant x \leqslant a \\ 0 & \text { otherwise } \end{cases}$$ where \(a\) and \(c\) are constants. State a suitable value for \(a\). (There is no need to evaluate \(c\).)

4 \includegraphics[max width=\textwidth, alt={}, center]{332f0909-c192-40f7-88b7-7cfec2db2eef-06_428_773_260_685} The time, \(X\) minutes, taken by a large number of runners to complete a certain race has probability density function f 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, as shown in the diagram.

1. Without calculation, explain how you can tell that there were more runners whose times were below 7.5 minutes than above 7.5 minutes.
2. Show that \(k = 10\).
3. Find \(\mathrm { E } ( X )\).
4. Find \(\operatorname { Var } ( X )\).

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 \(\mathrm { P } ( 0.4 < X < 2 )\).