5.03a Continuous random variables: pdf and cdf

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 \(\operatorname { Var } ( X )\).

5 \includegraphics[max width=\textwidth, alt={}, center]{c06524f0-a981-48a6-9af0-c4a3474396b3-06_394_723_258_705} The diagram shows the graph of the probability density function, f , of a random variable \(X\) which takes values between 0 and \(a\) only. It is given that \(\mathrm { P } ( X < 1 ) = 0.25\).

1. Find, in any order,
   1. \(\mathrm { P } ( X < 2 )\),
   2. the value of \(a\),
   3. \(\mathrm { f } ( x )\).
   4. Find the median of \(X\).

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 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 \(m > 0.5\).

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 \includegraphics[max width=\textwidth, alt={}, center]{f45a7c6f-ebb4-43a2-8751-11aab3561d3c-08_490_1195_255_475} The diagram shows the graph of the probability density function of a random variable \(X\) that takes values between - 1 and 3 only. It is given that the graph is symmetrical about the line \(x = 1\). Between \(x = - 1\) and \(x = 3\) the graph is a quadratic curve. The random variable \(S\) is such that \(\mathrm { E } ( S ) = 2 \times \mathrm { E } ( X )\) and \(\operatorname { Var } ( S ) = \operatorname { Var } ( X )\).

1. On the grid below, sketch a quadratic graph for the probability density function of \(S\). \includegraphics[max width=\textwidth, alt={}, center]{f45a7c6f-ebb4-43a2-8751-11aab3561d3c-08_490_1191_1169_479} The random variable \(T\) is such that \(\mathrm { E } ( T ) = \mathrm { E } ( X )\) and \(\operatorname { Var } ( T ) = \frac { 1 } { 4 } \operatorname { Var } ( X )\).
2. On the grid below, sketch a quadratic graph for the probability density function of \(T\). \includegraphics[max width=\textwidth, alt={}, center]{f45a7c6f-ebb4-43a2-8751-11aab3561d3c-08_488_1187_1996_479} It is now given that $$f ( x ) = \begin{cases} \frac { 3 } { 32 } \left( 3 + 2 x - x ^ { 2 } \right) & - 1 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{cases}$$
3. Given that \(\mathrm { P } ( 1 - a < X < 1 + a ) = 0.5\), show that \(a ^ { 3 } - 12 a + 8 = 0\).
4. Hence verify that \(0.69 < a < 0.70\).

6 \includegraphics[max width=\textwidth, alt={}, center]{8f9e5f25-05c1-4a4e-9094-2a1b42416588-08_490_1195_255_475} The diagram shows the graph of the probability density function of a random variable \(X\) that takes values between - 1 and 3 only. It is given that the graph is symmetrical about the line \(x = 1\). Between \(x = - 1\) and \(x = 3\) the graph is a quadratic curve. The random variable \(S\) is such that \(\mathrm { E } ( S ) = 2 \times \mathrm { E } ( X )\) and \(\operatorname { Var } ( S ) = \operatorname { Var } ( X )\).

1. On the grid below, sketch a quadratic graph for the probability density function of \(S\). \includegraphics[max width=\textwidth, alt={}, center]{8f9e5f25-05c1-4a4e-9094-2a1b42416588-08_490_1191_1169_479} The random variable \(T\) is such that \(\mathrm { E } ( T ) = \mathrm { E } ( X )\) and \(\operatorname { Var } ( T ) = \frac { 1 } { 4 } \operatorname { Var } ( X )\).
2. On the grid below, sketch a quadratic graph for the probability density function of \(T\). \includegraphics[max width=\textwidth, alt={}, center]{8f9e5f25-05c1-4a4e-9094-2a1b42416588-08_488_1187_1996_479} It is now given that $$f ( x ) = \begin{cases} \frac { 3 } { 32 } \left( 3 + 2 x - x ^ { 2 } \right) & - 1 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{cases}$$
3. Given that \(\mathrm { P } ( 1 - a < X < 1 + a ) = 0.5\), show that \(a ^ { 3 } - 12 a + 8 = 0\).
4. Hence verify that \(0.69 < a < 0.70\).

6 A continuous random variable \(X\) takes values from 0 to 6 only and has a probability distribution that is symmetrical. Two values, \(a\) and \(b\), of \(X\) are such that \(\mathrm { P } ( a < X < b ) = p\) and \(\mathrm { P } ( b < X < 3 ) = \frac { 13 } { 10 } p\), where \(p\) is a positive constant.

1. Show that \(p \leqslant \frac { 5 } { 23 }\).
2. Find \(\mathrm { P } ( b < X < 6 - a )\) in terms of \(p\).
   It is now given that the probability density function of \(X\) is f , where $$f ( x ) = \begin{cases} \frac { 1 } { 36 } \left( 6 x - x ^ { 2 } \right) & 0 \leqslant x \leqslant 6 \\ 0 & \text { otherwise } \end{cases}$$
3. Given that \(b = 2\) and \(p = \frac { 5 } { 27 }\), find the value of \(a\).

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

1. It is given that \(\mathrm { E } ( X ) = \ln 2\). Show that \(b = 2 a\).
2. Show that \(a = \frac { 1 } { 2 }\).
3. Find the median of \(X\).

4 A random variable \(X\) has probability density function f defined by $$f ( x ) = \begin{cases} \frac { a } { x ^ { 2 } } - \frac { 18 } { x ^ { 3 } } & 2 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{cases}$$ where \(a\) is a constant.

1. Show that \(a = \frac { 27 } { 2 }\).
2. Show that \(\mathrm { E } ( X ) = \frac { 27 } { 2 } \ln \frac { 3 } { 2 } - 3\).

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 \(\mathrm { P } ( X > 2 )\).

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

1. Sketch the probability density function of \(X\).
2. Show that the mean, \(\mu\), of \(X\) is 1.6875 .
3. Show that the standard deviation, \(\sigma\), of \(X\) is 0.2288 , correct to 4 decimal places.
4. Find \(\mathrm { P } ( 1 \leqslant X \leqslant \mu + \sigma )\).

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.

1 \includegraphics[max width=\textwidth, alt={}, center]{29ab8740-9fad-4596-b34f-c424b5464858-2_453_1258_251_447} Fred arrives at random times on a station platform. The times in minutes he has to wait for the next train are modelled by the continuous random variable for which the probability density function f is shown above.

1. State the value of \(k\).
2. Explain briefly what this graph tells you about the arrival times of trains.

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.

1 \includegraphics[max width=\textwidth, alt={}, center]{f73b303b-eb56-4c35-aa3e-388e3a3e8acd-2_453_1258_251_447} Fred arrives at random times on a station platform. The times in minutes he has to wait for the next train are modelled by the continuous random variable for which the probability density function f is shown above.

1. State the value of \(k\).
2. Explain briefly what this graph tells you about the arrival times of trains.

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 \(\operatorname { Var } ( X )\).