5.03b Solve problems: using pdf

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 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 \(X\).

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.

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.

6 The distance travelled, in kilometres, by a Grippo brake pad before it needs to be replaced is modelled by \(10000 X\), where \(X\) is a random variable having the probability density function $$f ( x ) = \begin{cases} - k \left( x ^ { 2 } - 5 x + 6 \right) & 2 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{cases}$$ The graph of \(y = \mathrm { f } ( x )\) is shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{c1dcf0f5-e971-4afd-81ca-4d860732825c-3_439_1100_580_520}

1. Show that \(k = 6\).
2. State the value of \(\mathrm { E } ( X )\) and find \(\operatorname { Var } ( X )\).
3. Sami fits four new Grippo brake pads on his car. Find the probability that at least one of these brake pads will need to be replaced after travelling less than 22000 km .

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

6 The time in minutes taken by people to read a certain booklet is modelled by the random variable \(T\) with probability density function given by $$f ( t ) = \begin{cases} \frac { 1 } { 2 \sqrt { } t } & 4 \leqslant t \leqslant 9 \\ 0 & \text { otherwise } \end{cases}$$

1. Find the time within which \(90 \%\) of people finish reading the booklet.
2. Find \(\mathrm { E } ( T )\) and \(\operatorname { Var } ( T )\).

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

1. Find \(\mathrm { E } ( X )\).
2. Find \(\mathrm { P } ( X < \mathrm { E } ( X ) )\).
3. Hence explain whether the mean of \(X\) is less than, equal to or greater than the median of \(X\).

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 \(a\).

5 The lifetime, \(X\) years, of a certain type of battery 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. State what the value of \(a\) represents in this context.
2. Show that \(k = \frac { a } { a - 1 }\).
3. Experience has shown that the longest that any battery of this type lasts is 2.5 years. Find the mean lifetime of batteries of this type.

1 \includegraphics[max width=\textwidth, alt={}, center]{cfffe79d-91c9-48b8-a3e6-887d7891441d-2_478_691_260_724} The random variable \(X\) has probability density function, f , as shown in the diagram, where \(a\) is a constant. Find the value of \(a\) and hence show that \(\mathrm { E } ( X ) = 0.943\) correct to 3 significant figures. [5]