Continuous Probability Distributions and Random Variables

1 The continuous random variable \(X\) has probability density function given by $$\mathrm { f } ( x ) = \begin{cases} a & 0 \leqslant x \leqslant 1 , \\ \frac { a } { x ^ { 2 } } & x > 1 , \\ 0 & \text { otherwise. } \end{cases}$$ Find the value of the constant \(a\).

1 The random variable \(T\) has probability density defined by $$\mathrm { f } ( t ) = \left\{ \begin{array} { c c } \frac { t } { 8 } & 0 \leq t \leq k \\ 0 & \text { otherwise } \end{array} \right.$$ Find the value of \(k\) [0pt] [1 mark] $$\begin{array} { l l l l } \frac { 1 } { 16 } & \frac { 1 } { 4 } & 4 & 16 \end{array}$$

5 \begin{figure}[h] \end{figure} The random variable \(X\) has probability density function \(\mathrm { f } ( x )\) and Figure 1 shows a sketch of \(\mathrm { f } ( x )\) where $$f ( x ) = \left\{ \begin{array} { c c } k ( 1 - \cos x ) & 0 \leqslant x \leqslant 2 \pi \\ 0 & \text { otherwise } \end{array} \right.$$

1. Show that \(k = \frac { 1 } { 2 \pi }\) The random variable \(Y \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)\) and \(\mathrm { E } ( Y ) = \mathrm { E } ( X )\) The probability density function of \(Y\) is \(g ( y )\), where $$g ( y ) = \frac { 1 } { \sigma \sqrt { 2 \pi } } e ^ { - \frac { 1 } { 2 } \left( \frac { y - \mu } { \sigma } \right) ^ { 2 } } \quad - \infty < y < \infty$$ Given that \(\mathrm { g } ( \mu ) = \mathrm { f } ( \mu )\)
2. find the exact value of \(\sigma\)
3. Calculate the error in using \(\mathrm { P } \left( \frac { \pi } { 2 } < Y < \frac { 3 \pi } { 2 } \right)\) as an approximation to \(\mathrm { P } \left( \frac { \pi } { 2 } < X < \frac { 3 \pi } { 2 } \right)\)

1 \includegraphics[max width=\textwidth, alt={}, center]{879cb813-2380-47a7-bd96-cad0a74d0b4d-2_369_531_255_806} The diagram shows the graph of the probability density function, f , of a random variable \(X\). Find the median of \(X\).

1 \includegraphics[max width=\textwidth, alt={}, center]{879cb813-2380-47a7-bd96-cad0a74d0b4d-2_369_531_255_806} The diagram shows the graph of the probability density function, f , of a random variable \(X\). Find the median of \(X\).

6 The object distance, \(U \mathrm {~cm}\), and the image distance, \(V \mathrm {~cm}\), for a convex lens of focal length 40 cm are related by the lens law $$\frac { 1 } { U } + \frac { 1 } { V } = \frac { 1 } { 40 } .$$ The random variable \(U\) is uniformly distributed over the interval \(80 \leqslant u \leqslant 120\).

1. Show that the probability density function of \(V\) is given by $$f ( v ) = \begin{cases} \frac { 40 } { ( v - 40 ) ^ { 2 } } & 60 \leqslant v \leqslant 80 \\ 0 & \text { otherwise } \end{cases}$$
2. Find
   1. the median value of \(V\),
   2. the expected value of \(V\).

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

1. Find the distribution function of \(X\). [3]
2. Find the exact value of the interquartile range of \(X\). [5]

1. Lloyd regularly takes a break from work to go to the local cafe. The amount of time Lloyd waits to be served, in minutes, is modelled by the continuous random variable \(T\), having probability density function

$$f ( t ) = \left\{ \begin{array} { c c } \frac { t } { 120 } & 4 \leqslant t \leqslant 16 \\ 0 & \text { otherwise } \end{array} \right.$$

1. Show that the cumulative distribution function is given by $$\mathrm { F } ( t ) = \left\{ \begin{array} { c r } 0 & t < 4 \\ \frac { t ^ { 2 } } { 240 } - c & 4 \leqslant t \leqslant 16 \\ 1 & t > 16 \end{array} \right.$$ where the value of \(c\) is to be found.
2. Find the exact probability that the amount of time Lloyd waits to be served is between 5 and 10 minutes.
3. Find the median of \(T\).
4. Find the value of \(k\) such that $$\mathrm { P } ( T < k ) = \frac { 2 } { 3 } \mathrm { P } ( T > k )$$ giving your answer to 3 significant figures.

6 The continuous random variable \(X\) has probability density function f given by $$f ( x ) = \begin{cases} \frac { 3 } { 28 } \left( e ^ { \frac { 1 } { 2 } x } + 4 e ^ { - \frac { 1 } { 2 } x } \right) & 0 \leqslant x \leqslant 2 \ln 3 \\ 0 & \text { otherwise } \end{cases}$$

1. Find the cumulative distribution function of \(X\).
   The random variable \(Y\) is defined by \(Y = e ^ { \frac { 1 } { 2 } ( X ) }\).
2. Find the probability density function of \(Y\).
3. Find the 30th percentile of \(Y\).
4. Find \(\mathrm { E } \left( Y ^ { 4 } \right)\).
   If you use the following page to complete the answer to any question, the question number must be clearly shown.

The graph of the probability density function of a random variable \(X\) is symmetrical about the line \(x = 4\). Given that \(\text{P}(X < 5) = \frac{20}{39}\), find \(\text{P}(3 < X < 5)\). [2]

7. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the probability density function \(\mathrm { f } ( x )\) of the random variable \(X\). The part of the sketch from \(x = 0\) to \(x = 4\) consists of an isosceles triangle with maximum at ( \(2,0.5\) ).

1. Write down \(\mathrm { E } ( X )\). The probability density function \(\mathrm { f } ( x )\) can be written in the following form. $$f ( x ) = \begin{cases} a x & 0 \leqslant x < 2 \\ b - a x & 2 \leqslant x \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$
2. Find the values of the constants \(a\) and \(b\).
3. Show that \(\sigma\), the standard deviation of \(X\), is 0.816 to 3 decimal places.
4. Find the lower quartile of \(X\).
5. State, giving a reason, whether \(\mathrm { P } ( 2 - \sigma < X < 2 + \sigma )\) is more or less than 0.5

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 \(a\).

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

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

3 The monthly demand for a product, \(X\) thousand units, is modelled by the random variable \(X\) with probability density function given by $$f ( x ) = \begin{cases} a x & 0 \leqslant x \leqslant 1 \\ a ( x - 2 ) ^ { 2 } & 1 < x \leqslant 2 \\ 0 & \text { otherwise } \end{cases}$$ where \(a\) is a positive constant. Find

1. the value of \(a\),
2. the probability that the monthly demand is at most 1500 units,
3. the expected monthly demand.

8 The continuous random variable \(X\) has probability density function \(\mathrm { f } ( x )\) It is given that \(\mathrm { f } ( x ) = x ^ { 2 }\) for \(0 \leq x \leq 1\) It is also given that \(\mathrm { f } ( x )\) is a linear function for \(1 < x \leq \frac { 3 } { 2 }\) For all other values of \(x , \mathrm { f } ( x ) = 0\) A sketch of the graph of \(y = \mathrm { f } ( x )\) is shown below. \includegraphics[max width=\textwidth, alt={}, center]{c309e27b-5618-4f94-aecd-a55d8756ef03-12_821_1077_758_543} Show that \(\operatorname { Var } ( X ) = 0.0864\) correct to three significant figures. \includegraphics[max width=\textwidth, alt={}, center]{c309e27b-5618-4f94-aecd-a55d8756ef03-14_2491_1755_173_123} Additional page, if required. number Write the question numbers in the left-hand margin.

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]

3 \includegraphics[max width=\textwidth, alt={}, center]{9fca48da-82c3-4ce1-9e0c-93eb9a920f9d-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 \(\mathrm { E } ( X )\).

The continuous random variable \(X\) takes values in the interval \(0 \leqslant x \leqslant 3\) only. For \(0 \leqslant x \leqslant 3\) the graph of its probability density function f consists of two straight line segments meeting at the point \(( 1 , k )\), as shown in the diagram. Find \(k\) and hence show that the distribution function F is given by $$\mathrm { F } ( x ) = \begin{cases} 0 & x \leqslant 0 , \\ \frac { 1 } { 3 } x ^ { 2 } & 0 < x \leqslant 1 , \\ x - \frac { 1 } { 2 } - \frac { 1 } { 6 } x ^ { 2 } & 1 < x \leqslant 3 , \\ 1 & x > 3 . \end{cases}$$ The random variable \(Y\) is given by \(Y = X ^ { 2 }\). Find

1. the probability density function of \(Y\),
2. the median value of \(Y\).

1 Calculate the variance of the continuous random variable with probability density function given by $$f ( x ) = \begin{cases} \frac { 3 } { 37 } x ^ { 2 } & 3 \leqslant x \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$

1 Calculate the variance of the continuous random variable with probability density function given by $$f ( x ) = \begin{cases} \frac { 3 } { 37 } x ^ { 2 } & 3 \leqslant x \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$

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 \(\mathrm { P } ( T \leqslant b ) = \frac { 3 } { 4 }\).

1. The continuous random variable \(X\) has probability density function \(\mathrm { f } ( x )\) given by

$$f ( x ) = \left\{ \begin{array} { c c } 2 ( x - 2 ) & 2 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{array} \right.$$

1. Sketch \(\mathrm { f } ( x )\) for all values of \(x\).
2. Write down the mode of \(X\). Find
3. \(\mathrm { E } ( X )\),
4. the median of \(X\).
5. Comment on the skewness of this distribution. Give a reason for your answer.

6 The lengths of time, in years, that sales representatives for a certain company keep their company cars may be modelled by the distribution with probability density function \(\mathrm { f } ( x )\), where $$f ( x ) = \left\{ \begin{array} { c c } \frac { 4 } { 27 } x ^ { 2 } ( 3 - x ) & 0 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{array} \right.$$

1. Draw a sketch of this probability density function.
2. Calculate the mean and the mode of \(X\).
3. Comment briefly on the values obtained in part (ii) in relation to the sketch in part (i).
4. Show that the lower quartile \(\mathrm { Q } _ { 1 }\) of \(X\) satisfies the equation \(\mathrm { Q } _ { 1 } { } ^ { 4 } - 4 \mathrm { Q } _ { 1 } { } ^ { 3 } + 6.75 = 0\), and use an appropriate numerical method to find the value of \(\mathrm { Q } _ { 1 }\) correct to 2 decimal places, showing full details of your method.

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 \(X\).

1 Let \(X\) be a continuous random variable with probability density function given by $$f ( x ) = \begin{cases} \frac { 3 } { 4 } x ( 2 - x ) & 0 \leq x \leq 2 \\ 0 & \text { otherwise } \end{cases}$$ Find \(\mathrm { P } ( X = 1 )\) Circle your answer.
0 \(\frac { 1 } { 2 }\) \(\frac { 3 } { 4 }\) \(\frac { 27 } { 32 }\)

11 The length of time in minutes for which a particular geyser erupts is modelled by the continuous random variable \(T\) with cumulative distribution function given by \(\mathrm { F } ( t ) = \begin{cases} 0 & t \leqslant 2 , \\ k \left( 8 t ^ { 2 } - t ^ { 3 } - 24 \right) & 2 < t < 4 , \\ 1 & t \geqslant 4 , \end{cases}\) where \(k\) is a positive constant.

1. Show that \(k = \frac { 1 } { 40 }\).
2. Find the probability that a randomly selected eruption time lies between 2.5 and 3.5 minutes.
3. Show that the median \(m\) of the distribution satisfies the equation \(m ^ { 3 } - 8 m ^ { 2 } + 44 = 0\).
4. Verify that the median eruption time is 2.95 minutes, correct to 2 decimal places. The mean and standard deviation of \(T\) are denoted by \(\mu\) and \(\sigma\) respectively.
5. Find \(\mathrm { P } ( \mu - \sigma < T < \mu + \sigma )\).
6. Sketch the graph of the probability density function of \(T\).
7. A Normally distributed random variable \(X\) has the same mean and standard deviation as \(T\). By considering the shape of the Normal distribution, and without doing any calculations, explain whether \(\mathrm { P } ( \mu - \sigma < X < \mu + \sigma )\) will be greater than, equal to or less than the probability that you calculated in part (e).

1. The continuous random variable \(X\) has probability density function given by

The continuous random variable \(X\) has probability density function $$\text{f}(x) = \begin{cases} kx(5 - x), & 0 \leq x \leq 4, \\ 0, & \text{otherwise,} \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = \frac{3}{56}\). [3]
2. Find the cumulative distribution function F(\(x\)) for all values of \(x\). [4]
3. Evaluate E(\(X\)). [3]
4. Find the modal value of \(X\). [3]
5. Verify that the median value of \(X\) lies between 2.3 and 2.5. [3]
6. Comment on the skewness of \(X\). Justify your answer. [2]

3. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the probability density function \(\mathrm { f } ( x )\) of the random variable \(X\).
For \(0 \leqslant x \leqslant 3 , \mathrm { f } ( x )\) is represented by a curve \(O B\) with equation \(\mathrm { f } ( x ) = k x ^ { 2 }\), where \(k\) is a constant. For \(3 \leqslant x \leqslant a\), where \(a\) is a constant, \(\mathrm { f } ( x )\) is represented by a straight line passing through \(B\) and the point ( \(a , 0\) ). For all other values of \(x , \mathrm { f } ( x ) = 0\).
Given that the mode of \(X =\) the median of \(X\), find

1. the mode,
2. the value of \(k\),
3. the value of \(a\). Without calculating \(\mathrm { E } ( X )\) and with reference to the skewness of the distribution
4. state, giving your reason, whether \(\mathrm { E } ( X ) < 3 , \mathrm { E } ( X ) = 3\) or \(\mathrm { E } ( X ) > 3\).

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

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

1. Show that \(| a | \leqslant \frac { 1 } { 2 }\).
2. Find \(\mathrm { E } ( X )\) in terms of \(a\).
3. Construct an unbiased estimator \(T _ { 1 }\) of \(a\) based on one observation \(X _ { 1 }\) of \(X\).
4. A second observation \(X _ { 2 }\) is taken. Show that \(T _ { 2 }\), where \(T _ { 2 } = \frac { 3 } { 8 } \left( X _ { 1 } + X _ { 2 } \right)\), is also an unbiased estimator of a.
5. Given that \(\operatorname { Var } ( X ) = \sigma ^ { 2 }\), determine which of \(T _ { 1 }\) and \(T _ { 2 }\) is the better estimator.

5. The continuous random variable \(X\) has probability density function \(\mathrm { f } ( x )\) given by $$f ( x ) = \left\{ \begin{array} { c c } k \left( x ^ { 2 } + a \right) & - 1 < x \leqslant 2 \\ 3 k & 2 < x \leqslant 3 \\ 0 & \text { otherwise } \end{array} \right.$$ where \(k\) and \(a\) are constants.
Given that \(\mathrm { E } ( X ) = \frac { 17 } { 12 }\)

1. find the value of \(k\) and the value of \(a\)
2. Write down the mode of \(X\)

6 The continuous random variable \(X\) has the following probability density function: $$f ( x ) = \begin{cases} a + b x & 0 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{cases}$$ where \(a\) and \(b\) are constants.

1. Show that \(2 a + 2 b = 1\).
2. It is given that \(\mathrm { E } ( X ) = \frac { 11 } { 9 }\). Use this information to find a second equation connecting \(a\) and \(b\), and hence find the values of \(a\) and \(b\).
3. Determine whether the median of \(X\) is greater than, less than, or equal to \(\mathrm { E } ( X )\).

4 \includegraphics[max width=\textwidth, alt={}, center]{a9f9cf66-0734-4316-99ae-c57090d08135-08_353_1141_255_463} The diagram shows the continuous random variable \(X\) with probability density function f given by $$f ( x ) = \begin{cases} \frac { 1 } { 128 } \left( 4 a x - b x ^ { 3 } \right) & 0 \leqslant x \leqslant 4 \\ c & 4 \leqslant x \leqslant 6 \\ 0 & \text { otherwise } \end{cases}$$ where \(a , b\) and \(c\) are constants.
The upper quartile of \(X\) is equal to 4 .

1. Show that \(c = \frac { 1 } { 8 }\) and find the values of \(a\) and \(b\).
2. Find the exact value of the median of \(X\).
3. Find \(\mathrm { E } ( \sqrt { X } )\), giving your answer correct to 2 decimal places.

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

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

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

1. \(\mathrm { E } ( X )\),
2. \(\mathrm { P } ( X \geqslant \mathrm { E } ( X ) )\).

5 The diagram shows a graph of the probability density function of the random variable \(X\). \includegraphics[max width=\textwidth, alt={}, center]{313cd5ce-07ff-4781-a134-565b8b221145-05_574_1086_406_479} 5

1. State the mode of \(X\).
   5
2. Find the probability density function of \(X\).

7 The continuous random variable \(X\) has probability density function \(f ( x ) = \begin{cases} k x ^ { n } & 0 \leqslant x \leqslant 1 , \\ 0 & \text { otherwise, } \end{cases}\) where \(k\) is a constant and \(n\) is a parameter whose value is positive. It is given that the median of \(X\) is 0.8816 correct to 4 decimal places. Ten independent observations of \(X\) are obtained. Find the expected number of observations that are less than 0.8 .

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 \(X\).

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 \(X\).

A student states that \(\int_0^{\pi/2} \frac{\cos x + \sin x}{\cos x - \sin x} \, dx\) is not an improper integral because \(\frac{\cos x + \sin x}{\cos x - \sin x}\) is defined at both \(x = 0\) and \(x = \frac{\pi}{2}\) Assess the validity of the student's argument. [2 marks]

The continuous random variable X has cumulative distribution function F(x) given by $$\text{F}(x) = \begin{cases} 0, & x < 1 \\ \frac{1}{2}(-x^3 + 6x^2 - 5), & 1 \leq x \leq 4 \\ 1, & x > 4 \end{cases}$$

1. Find the probability density function f(x). [3]
2. Find the mode of X. [2]
3. Sketch f(x) for all values of x. [3]
4. Find the mean \(\mu\) of X. [3]
5. Show that F(\(\mu\)) > 0.5. [1]
6. Show that the median of X lies between the mode and the mean. [2]

4

1. \(\text { Find } \mathrm { P } ( X > 1 )\) [0pt] [3 marks]
   4
2. 
   [0pt] [3 marks]
   □
   4
3. Find \(\mathrm { E } \left( 2 X ^ { - 1 } - 3 \right)\)

1. Show that \(b = \frac { a } { a - 1 }\).
2. Given that the median of \(X\) is \(\frac { 3 } { 2 }\), find the values of \(a\) and \(b\).
3. Use your values of \(a\) and \(b\) from part (ii) to find \(\mathrm { E } ( X )\).

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

1. Find the distribution function of \(X\). The random variable \(Y\) is defined by \(Y = X ^ { 3 }\). Find
2. the probability density function of \(Y\),
3. the value of \(k\) for which \(\mathrm { P } ( Y \geqslant k ) = \frac { 7 } { 12 }\).

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.

7 The function \(\mathrm { f } ( x )\) is defined by $$f ( x ) = \begin{cases} \frac { 1 } { 4 } x \left( 4 - x ^ { 2 } \right) & 0 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{cases}$$

1. Show that \(\mathrm { f } ( x )\) satisfies the conditions for a probability density function.
2. Find the value of \(a\) such that \(\mathrm { P } ( X < a ) = \frac { 15 } { 16 }\).

1. Given that \(\mathrm { P } ( X \leqslant 2 ) = \frac { 1 } { 3 }\), show that \(m = \frac { 1 } { 6 }\) and find the values of \(k\) and \(c\).
2. Find the exact numerical value of the interquartile range of \(X\).

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 \(\mathrm { P } ( X < a ) = 0.75\).

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 The lifetime, \(T\) years, of a mortgage may be modelled by the random variable \(T\) with probability density function \(\mathrm { f } ( t )\), where $$\mathrm { f } ( t ) = \begin{cases} k \sin \left( \frac { 3 } { 32 } t \right) & 0 \leqslant t \leqslant 8 \pi \\ 0 & \text { otherwise } \end{cases}$$

1. Show that \(k = \frac { 3 } { 32 } ( 2 - \sqrt { 2 } )\).
2. Sketch the graph of \(\mathrm { f } ( t )\) and state the mode.

6. The continuous random variable \(X\) has (cumulative) distribution function given by $$F ( x ) = \left\{ \begin{array} { c c } 0 & x < 1 \\ 1 - \frac { 1 } { x ^ { 4 } } & x \geq 1 \end{array} \right.$$ a. Show that the probability density function of \(Y\), where \(Y = \frac { 1 } { X ^ { 2 } }\), is given by $$g ( y ) = \left\{ \begin{array} { c c } 2 y & 0 < y \leq 1 \\ 0 & \text { otherwise } \end{array} \right.$$ b. Find \(\mathrm { E } ( \sqrt [ 3 ] { Y } )\).

A continuous random variable \(X\) has probability density function f(\(x\)) where $$f(x) = \begin{cases} kx^n & 0 \leq x \leq 1 \\ 0 & \text{otherwise} \end{cases}$$ where \(k\) and \(n\) are positive integers.

1. Find \(k\) in terms of \(n\). [3]
2. Find E(\(X\)) in terms of \(n\). [3]
3. Find E(\(X^2\)) in terms of \(n\). [2]

Given that \(n = 2\)

1. find Var(3\(X\)). [3]

4 The continuous random variable \(X\) has probability density function $$\mathrm { f } ( x ) = \begin{cases} \frac { k } { x ^ { n } } & x \geqslant 1 \\ 0 & \text { otherwise } \end{cases}$$ where \(n\) and \(k\) are constants and \(n\) is an integer greater than 1 .

1. Find \(k\) in terms of \(n\).
2. 1. When \(n = 4\), find the cumulative distribution function of \(X\).
   2. Hence determine \(\mathrm { P } ( X > 7 \mid X > 5 )\) when \(n = 4\).
3. Determine the values of \(n\) for which \(\operatorname { Var } ( X )\) is not defined.

A continuous random variable \(X\) has the probability density function f(\(x\)) shown in Figure 1. \includegraphics{figure_1} Figure 1

1. Show that f(\(x\)) = \(4 - 8x\) for \(0 \leqslant x \leqslant 0.5\) and specify f(\(x\)) for all real values of \(x\). [4]
2. Find the cumulative distribution function F(\(x\)). [4]
3. Find the median of \(X\). [3]
4. Write down the mode of \(X\). [1]
5. State, with a reason, the skewness of \(X\). [1]

3 The continuous random variable \(X\) has probability density function f given by $$f ( x ) = \begin{cases} \frac { 3 } { 16 } ( 2 - \sqrt { x } ) & 0 \leqslant x < 1 \\ \frac { 3 } { 16 \sqrt { x } } & 1 \leqslant x \leqslant 9 \\ 0 & \text { otherwise } \end{cases}$$

1. Find \(\mathrm { E } ( X )\).
   The random variable \(Y\) is such that \(Y = \sqrt { X }\).
2. Find the probability density function of \(Y\).