5.03a Continuous random variables: pdf and cdf

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

4 \includegraphics[max width=\textwidth, alt={}, center]{fb305858-2d96-4a5d-b1a9-a965c248fb8d-07_316_984_260_577} The diagram shows the probability density function, \(\mathrm { f } ( x )\), of a random variable \(X\). For \(0 \leqslant x \leqslant a\), \(\mathrm { f } ( x ) = k\); elsewhere \(\mathrm { f } ( x ) = 0\).

1. Express \(k\) in terms of \(a\).
2. Given that \(\operatorname { Var } ( X ) = 3\), find \(a\).

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 \includegraphics[max width=\textwidth, alt={}, center]{937c15d2-fb12-4af8-96d3-c54c81d771ba-07_316_984_260_577} The diagram shows the probability density function, \(\mathrm { f } ( x )\), of a random variable \(X\). For \(0 \leqslant x \leqslant a\), \(\mathrm { f } ( x ) = k\); elsewhere \(\mathrm { f } ( x ) = 0\).

1. Express \(k\) in terms of \(a\).
2. Given that \(\operatorname { Var } ( X ) = 3\), find \(a\).

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 \(\mathrm { P } ( 1 \leqslant x \leqslant 3 )\).

7 \includegraphics[max width=\textwidth, alt={}, center]{a93e5413-6ad8-4957-8efd-470cf79792e2-12_428_693_260_724} A random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} ( \sqrt { } 2 ) \cos x & 0 \leqslant x \leqslant \frac { 1 } { 4 } \pi \\ 0 & \text { otherwise } \end{cases}$$ as shown in the diagram.

1. Find \(\mathrm { P } \left( X > \frac { 1 } { 6 } \pi \right)\).
2. Find the median of \(X\).
3. Find \(\mathrm { E } ( X )\).
   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 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.

7 The time in hours taken for clothes to dry can be modelled by the continuous random variable with probability density function given by $$f ( t ) = \begin{cases} k \sqrt { } t & 1 \leqslant t \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = \frac { 3 } { 14 }\).
2. Find the mean time taken for clothes to dry.
3. Find the median time taken for clothes to dry.
4. Find the probability that the time taken for clothes to dry is between the mean time and the median time.

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

1. Show that \(k = \sqrt { } 2\).
2. Find \(\mathrm { P } ( X > 0.4 )\).
3. Find the upper quartile of \(X\).
4. Find the probability that exactly 3 out of 5 random observations of \(X\) have values greater than the upper quartile.

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

1. Show that the value of \(k\) is \(\frac { 32 } { 9 }\).
2. Find \(\mathrm { E } ( X )\).
3. Is the median less than or greater than 1.5? Justify your answer numerically.

4 \includegraphics[max width=\textwidth, alt={}, center]{c7cbd61b-9a62-494a-b595-f624ec5c0bea-2_351_561_1562_794} The diagram shows the graph of the probability density function, f , of a random variable \(X\) which takes values between 0 and 2 only.

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

4 \includegraphics[max width=\textwidth, alt={}, center]{0784d885-5710-4eb4-8cf8-2582122bf7ed-2_351_554_1562_794} The diagram shows the graph of the probability density function, f , of a random variable \(X\) which takes values between 0 and 2 only.

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

5 A continuous random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} \frac { 1 } { 6 } x & 2 \leqslant x \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$

1. Find \(\mathrm { E } ( X )\).
2. Find the median of \(X\).
3. Two independent values of \(X\) are chosen at random. Find the probability that both these values are greater than 3 .

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

1. Show that \(k = \frac { 1 } { \ln 2 }\).
2. Find \(a\) such that \(\mathrm { P } ( X < a ) = 0.75\).

6 Darts are thrown at random at a circular board. The darts hit the board at distances \(X\) centimetres from the centre, where \(X\) is a random variable with probability density function given by $$f ( x ) = \begin{cases} \frac { 2 } { a ^ { 2 } } x & 0 \leqslant x \leqslant a \\ 0 & \text { otherwise } \end{cases}$$ where \(a\) is a positive constant.

1. Verify that f is a probability density function whatever the value of \(a\). It is now given that \(\mathrm { E } ( X ) = 8\).
2. Find the value of \(a\).
3. Find the probability that a dart lands more than 6 cm from the centre of the board.

5 The volume, in \(\mathrm { cm } ^ { 3 }\), of liquid left in a glass by people when they have finished drinking all they want is modelled by the random variable \(X\) with probability density function given by $$f ( x ) = \begin{cases} k ( x - 2 ) ^ { 2 } & 0 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = \frac { 3 } { 8 }\).
2. 20\% of people leave at least \(d \mathrm {~cm} ^ { 3 }\) of liquid in a glass. Find \(d\).
3. Find \(\mathrm { E } ( X )\).