5.03b Solve problems: using pdf

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.

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 \(a\).

5 \includegraphics[max width=\textwidth, alt={}, center]{b054d0a0-01b6-4785-807c-851551b90544-06_382_743_260_699} The diagram shows the probability density function, f , of a random variable \(X\), in terms of the constants \(a\) and \(b\).

1. Find \(b\) in terms of \(a\).
2. Show that \(\mathrm { f } ( x ) = \frac { 2 } { a } - \frac { 2 } { a ^ { 2 } } x\).
3. Given that \(\mathrm { E } ( X ) = 0.5\), find \(a\).

4

1. 
   
   [diagram]
   The diagram shows the graph of the probability density function, f , of a random variable \(X\), where \(a\) is a constant greater than 0.5 . The graph between \(x = 0\) and \(x = a\) is a straight line parallel to the \(x\)-axis.
   1. Find \(\mathrm { P } ( X < 0.5 )\) in terms of \(a\).
   2. Find \(\mathrm { E } ( X )\) in terms of \(a\).
   3. Show that \(\operatorname { Var } ( X ) = \frac { 1 } { 12 } a ^ { 2 }\).
2. A random variable \(T\) has probability density function given by $$\operatorname { g } ( t ) = \begin{cases} \frac { 3 } { 2 ( t - 1 ) ^ { 2 } } & 2 \leqslant t \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$ Find the value of \(b\) such that \(\mathrm { P } ( T \leqslant b ) = \frac { 3 } { 4 }\).

6 A function f is defined by $$f ( x ) = \begin{cases} \frac { 3 x ^ { 2 } } { a ^ { 3 } } & 0 \leqslant x \leqslant a \\ 0 & \text { otherwise } \end{cases}$$ where \(a\) is a constant.

1. Show that f is a probability density function for all positive values of \(a\).
   The random variable \(X\) has probability density function f and the median of \(X\) is 2 .
2. Show that \(a = 2.52\), correct to 3 significant figures.
3. Find \(\mathrm { E } ( X )\).

6 The probability density function, f, of a random variable \(X\) is given by $$f ( x ) = \begin{cases} k \left( 6 x - x ^ { 2 } \right) & 0 \leqslant x \leqslant 6 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.
State the value of \(\mathrm { E } ( X )\) and show that \(\operatorname { Var } ( X ) = \frac { 9 } { 5 }\).

3 \includegraphics[max width=\textwidth, alt={}, center]{189bcf7b-279f-457b-8232-ace7f0c9797f-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 \(\mathrm { E } ( X )\).

7

1. \includegraphics[max width=\textwidth, alt={}, center]{3f1a0c67-03a4-4b4f-99c0-4336ba7d56b0-3_255_643_264_790} 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 \(\mathrm { P } ( X > 2 )\).
2. A random variable, \(W\), has probability density function given by $$\mathrm { g } ( w ) = \begin{cases} a w & 0 \leqslant w \leqslant b \\ 0 & \text { otherwise } \end{cases}$$ where \(a\) and \(b\) are constants. Given that the median of \(W\) is 2 , find \(a\) and \(b\).

5

1. \includegraphics[max width=\textwidth, alt={}, center]{61ba010c-d6a2-4c19-9998-0ae048244a32-06_292_517_264_338} \includegraphics[max width=\textwidth, alt={}, center]{61ba010c-d6a2-4c19-9998-0ae048244a32-06_289_518_264_858} \includegraphics[max width=\textwidth, alt={}, center]{61ba010c-d6a2-4c19-9998-0ae048244a32-06_273_510_365_1377} The diagram shows the graphs of three functions, \(f _ { 1 } , f _ { 2 }\) and \(f _ { 3 }\). The function \(f _ { 1 }\) is a probability density function.
   1. State the value of \(k\).
   2. For each of the functions \(\mathrm { f } _ { 2 }\) and \(\mathrm { f } _ { 3 }\), state why it cannot be a probability density function.
2. The probability density function g is defined by $$g ( x ) = \begin{cases} 6 \left( a ^ { 2 } - x ^ { 2 } \right) & - a \leqslant x \leqslant a \\ 0 & \text { otherwise } \end{cases}$$ where \(a\) is a constant.
   1. Show that \(a = \frac { 1 } { 2 }\).
   2. State the value of \(\mathrm { E } ( X )\).
   3. Find \(\operatorname { Var } ( X )\).

6 The graph of the probability density function f of a random variable \(X\) is symmetrical about the line \(x = 2\). It is given that \(\mathrm { P } ( 2 < X < 5 ) = \frac { 117 } { 256 }\).

1. Using only this information show that \(\mathrm { P } ( X > - 1 ) = \frac { 245 } { 256 }\).
   It is now given that, for \(x\) in a suitable domain, $$f ( x ) = k \left( 12 + 4 x - x ^ { 2 } \right) , \text { where } k \text { is a constant. }$$
2. Find the value of \(k\).
3. A different random variable \(X\) has probability density function \(\mathbf { g } ( x ) = \frac { 2 } { 9 } \left( 2 + x - x ^ { 2 } \right)\). The domain of \(X\) is all values of \(x\) for which \(\mathrm { g } ( x ) \geqslant 0\). Find \(\operatorname { Var } ( X )\). \includegraphics[max width=\textwidth, alt={}, center]{ff3433b0-baab-45e3-845e-56a794739bba-11_63_1547_447_347}

3 \includegraphics[max width=\textwidth, alt={}, center]{ec7cab36-683b-4022-9cac-fb3b4e64778a-04_332_1100_260_520} A random variable \(X\) takes values between 0 and 3 only and has probability density function as shown in the diagram, where \(c\) is a constant.

1. Show that \(c = \frac { 2 } { 3 }\).
2. Find \(\mathrm { P } ( X > 2 )\).
3. Calculate \(\mathrm { E } ( X )\).

4 The random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} \frac { k } { \sqrt { } x } & 0 < x \leqslant a \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) and \(a\) are constants. It is given that \(\mathrm { E } ( X ) = 3\).

1. Find the value of \(a\) and show that \(k = \frac { 1 } { 6 }\).
2. Find the median of \(X\).

6 The random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} k x ^ { - 1 } & 2 \leqslant x \leqslant 6 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = \frac { 1 } { \ln 3 }\).
2. Show that \(\mathrm { E } ( X ) = 3.64\), correct to 3 significant figures.
3. Given that the median of \(X\) is \(m\), find \(\mathrm { P } ( m < X < \mathrm { E } ( X ) )\).

4 The time, \(X\) hours, taken by a large number of runners to complete a race is modelled by the probability density function given by $$f ( x ) = \begin{cases} \frac { k } { ( x + 1 ) ^ { 2 } } & 0 \leqslant x \leqslant a \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) and \(a\) are constants.

1. Show that \(k = \frac { a + 1 } { a }\).
2. State what the constant \(a\) represents in this context.
   Three quarters of the runners take half an hour or less to complete the race.
3. Find the value of \(a\).

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

1. Find \(a\).
2. Show that \(\mathrm { E } ( X ) = \frac { 4 } { 3 }\).
   The median of \(X\) is denoted by \(m\).
3. Find \(\mathrm { P } ( \mathrm { E } ( X ) < X < m )\).

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

1. Show that \(k = \frac { 2 } { 9 }\).
2. Find \(\mathrm { P } ( 1 \leqslant X \leqslant 2 )\).
3. Find \(\operatorname { Var } ( X )\).

3 \includegraphics[max width=\textwidth, alt={}, center]{4a5f9f7e-b045-4c6f-8bda-6c4067668da2-04_332_1100_260_520} A random variable \(X\) takes values between 0 and 3 only and has probability density function as shown in the diagram, where \(c\) is a constant.

1. Show that \(c = \frac { 2 } { 3 }\).
2. Find \(\mathrm { P } ( X > 2 )\).
3. Calculate \(\mathrm { E } ( X )\).

4 A random variable \(X\) has probability density function given by $$\mathrm { f } ( x ) = \begin{cases} k ( 3 - x ) & 1 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = \frac { 2 } { 3 }\).
2. Find the median of \(X\).

3 The continuous random variable \(X\) has probability density function f given by $$f ( x ) = \begin{cases} \frac { 1 } { 5 } x & 0 \leqslant x < 2 \\ \frac { 2 } { 15 } ( 5 - x ) & 2 \leqslant x \leqslant 5 \\ 0 & \text { otherwise } \end{cases}$$

1. Find the cumulative distribution function of \(X\).
2. Find the median value of \(X\).
3. Find \(\mathrm { E } \left( X ^ { 2 } \right)\).
4. Find \(\mathrm { P } ( 1 \leqslant x \leqslant 3 )\).

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]{2fefee17-50bb-4375-80f6-7e4bc2606492-04_405_789_260_676} 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 \(\mathrm { P } ( p < X < 1 ) = 0.25\).

7 The lifetime, \(x\) years, of the power light on a freezer, which is left on continuously, can be modelled by the continuous random variable with density function given by $$\mathrm { f } ( x ) = \begin{cases} k \mathrm { e } ^ { - 3 x } & x > 0 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = 3\).
2. Find the lower quartile.
3. Find the mean lifetime.

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

1. \(\mathrm { P } ( X > 0.5 )\),
2. the mean and variance of \(X\).

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

1. Show that the value of \(a\) is \(\frac { 1 } { 2 }\).
2. Find \(\mathrm { P } ( X > 1.8 )\).
3. Find \(\mathrm { E } ( X )\).

7 At a town centre car park the length of stay in hours is denoted by the random variable \(X\), which has probability density function given by $$f ( x ) = \begin{cases} k x ^ { - \frac { 3 } { 2 } } & 1 \leqslant x \leqslant 9 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.

1. Interpret the inequalities \(1 \leqslant x \leqslant 9\) in the definition of \(\mathrm { f } ( x )\) in the context of the question.
2. Show that \(k = \frac { 3 } { 4 }\).
3. Calculate the mean length of stay. The charge for a length of stay of \(x\) hours is \(\left( 1 - \mathrm { e } ^ { - x } \right)\) dollars.
4. Find the length of stay for the charge to be at least 0.75 dollars
5. Find the probability of the charge being at least 0.75 dollars.