5.03c Calculate mean/variance: by integration

9 The continuous random variable \(X\) has distribution function F given by $$\mathrm { F } ( x ) = \begin{cases} 0 & x < 2 , \\ \frac { 1 } { 8 } x - \frac { 1 } { 4 } & 2 \leqslant x \leqslant 10 , \\ 1 & x > 10 . \end{cases}$$ Find the value of \(k\) for which \(\mathrm { P } ( X \geqslant k ) = 0.6\). The random variable \(Y\) is defined by \(Y = 2 \ln X\). Find the distribution function of \(Y\). Find the probability density function of \(Y\) and sketch its graph.

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.

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.

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 .

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 \(\mathrm { P } ( m < X < 3 )\).

1 A random variable \(X\) has probability density function f , where $$f ( x ) = \begin{cases} \frac { 3 } { 2 } \left( 1 - x ^ { 2 } \right) & 0 \leqslant x \leqslant 1 \\ 0 & \text { otherwise } \end{cases}$$ Find \(\mathrm { E } ( X )\).

6 \includegraphics[max width=\textwidth, alt={}, center]{5dfbc896-528c-40a6-a296-8a0aae90add4-10_451_469_255_799} The diagram shows the graph of the probability density function, f , of a random variable \(X\). The graph is a quarter circle entirely in the first quadrant with centre \(( 0,0 )\) and radius \(a\), where \(a\) is a positive constant. Elsewhere \(\mathrm { f } ( x ) = 0\).

1. Show that \(a = \frac { 2 } { \sqrt { \pi } }\).
2. Show that \(\mathrm { f } ( x ) = \sqrt { \frac { 4 } { \pi } - x ^ { 2 } }\).
3. Show that \(\mathrm { E } ( X ) = \frac { 8 } { 3 \sqrt { \pi ^ { 3 } } }\).

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

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.

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

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.

4

1. \includegraphics[max width=\textwidth, alt={}, center]{7c9a87ac-69c6-4850-82aa-8235bba581e8-2_611_712_1466_358} \includegraphics[max width=\textwidth, alt={}, center]{7c9a87ac-69c6-4850-82aa-8235bba581e8-2_618_716_1464_1155} The diagrams show the graphs of two functions, \(g\) and \(h\). For each of the functions \(g\) and \(h\), give a reason why it cannot be a probability density function.
2. The distance, in kilometres, travelled in a given time by a cyclist is represented by the continuous random variable \(X\) with probability density function given by $$f ( x ) = \begin{cases} \frac { 30 } { x ^ { 2 } } & 10 \leqslant x \leqslant 15 \\ 0 & \text { otherwise } \end{cases}$$
   1. Show that \(\mathrm { E } ( X ) = 30 \ln 1.5\).
   2. Find the median of \(X\). Find also the probability that \(X\) lies between the median and the mean.

4 A random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} \frac { 1 } { 18 } \left( 9 - x ^ { 2 } \right) & 0 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{cases}$$

1. Find \(\mathrm { P } ( X < 1.2 )\).
2. Find \(\mathrm { E } ( X )\).
   The median of \(X\) is \(m\).
3. Show that \(m ^ { 3 } - 27 m + 27 = 0\).

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.
   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.

6 The time, \(X\) hours, taken by a large number of people to complete a challenge is modelled by the probability density function given by $$f ( x ) = \left\{ \begin{array} { c l } \frac { 1 } { x ^ { 2 } } & a \leqslant x \leqslant b \\ 0 & \text { otherwise } \end{array} \right.$$ where \(a\) and \(b\) are constants.

1. State what the constants \(a\) and \(b\) represent in this context.
2. Show that \(a = \frac { b } { b + 1 }\).
   It is given that \(\mathrm { E } ( X ) = \ln 3\).
3. Show that \(b = 2\) and find the value of \(a\). \includegraphics[max width=\textwidth, alt={}, center]{9ac74d4c-f5e0-4c5d-ab25-5692dfb06f0b-09_2726_35_97_20}
4. Find the median of \(X\).

5 \includegraphics[max width=\textwidth, alt={}, center]{43403c12-93e6-44e4-b15e-e3c4363be5f9-08_254_634_260_717} The diagram shows the graph of the probability density function, f , of a random variable \(X\), where $$f ( x ) = \begin{cases} \frac { 2 } { 9 } \left( 3 x - x ^ { 2 } \right) & 0 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{cases}$$

1. State the value of \(\mathrm { E } ( X )\) and find \(\operatorname { Var } ( X )\).
2. State the value of \(\mathrm { P } ( 1.5 \leqslant X \leqslant 4 )\).
3. Given that \(\mathrm { P } ( 1 \leqslant X \leqslant 2 ) = \frac { 13 } { 27 }\), find \(\mathrm { P } ( X > 2 )\).