5.03e Find cdf: by integration

1 The continuous random variable \(X\) has probability density function f given by $$f ( x ) = \begin{cases} \frac { 1 } { 6 } \left( x ^ { - \frac { 1 } { 3 } } - x ^ { - \frac { 2 } { 3 } } \right) & 1 \leqslant x \leqslant 27 \\ 0 & \text { otherwise } \end{cases}$$

1. Find the cumulative distribution function of \(X\).
   The random variable \(Y\) is defined by \(Y = X ^ { \frac { 1 } { 3 } }\).
2. Find the probability density function of \(Y\).
3. Find the exact value of the median of \(Y\).

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

1. Sketch the graph of f.
2. Show that \(k = \frac { 4 } { 33 }\). \includegraphics[max width=\textwidth, alt={}, center]{e2a45d19-7d48-4aa5-93f9-6ef90f99d7c4-09_2725_35_99_20}
3. Find the cumulative distribution function of \(X\).
4. Find the median value of \(X\).

4 The random variable \(X\) has probability density function f given by $$f ( x ) = \begin{cases} \frac { 1 } { 21 } ( x - 1 ) ^ { 2 } & 2 \leqslant x \leqslant 5 \\ 0 & \text { otherwise } \end{cases}$$

1. Find the cumulative distribution function of \(X\).
   The random variable \(Y\) is defined by \(Y = ( X - 1 ) ^ { 4 }\).
2. Find the probability density function of \(Y\). \includegraphics[max width=\textwidth, alt={}, center]{b9cbf607-4f40-41bb-8374-6b2c39f945ac-09_2725_35_99_20}
3. Find the median value of \(Y\).
4. Find \(\mathrm { E } ( Y )\).

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

1. Sketch the graph of f.
2. Show that \(k = \frac { 4 } { 33 }\). \includegraphics[max width=\textwidth, alt={}, center]{8b2a13d7-62f4-45a7-84c5-7d5bc870b8ce-09_2725_35_99_20}
3. Find the cumulative distribution function of \(X\).
4. Find the median value of \(X\).

6 The lifetime of a particular machine, in months, can be modelled by the random variable \(T\) with probability density function given by $$\mathrm { f } ( t ) = \begin{cases} \frac { 3 } { t ^ { 4 } } & t \geqslant 1 \\ 0 & \text { otherwise. } \end{cases}$$

1. Obtain the (cumulative) distribution function of \(T\).
2. Show that the probability density function of the random variable \(Y\), where \(Y = T ^ { 3 }\), is given by \(\mathrm { g } ( y ) = \frac { 1 } { y ^ { 2 } }\), for \(y \geqslant 1\).
3. Find \(\mathrm { E } ( \sqrt { Y } )\).

7 The continuous random variable \(T\) has probability density function given by $$f ( t ) = \begin{cases} 4 t ^ { 3 } & 0 < t \leqslant 1 \\ 0 & \text { otherwise } \end{cases}$$

1. Obtain the cumulative distribution function of \(T\).
2. Find the cumulative distribution function of \(H\), where \(H = \frac { 1 } { T ^ { 4 } }\), and hence show that the probability density function of \(H\) is given by \(\mathrm { g } ( h ) = \frac { 1 } { h ^ { 2 } }\) over an interval to be stated.
3. Find \(\mathrm { E } \left( 1 + 2 H ^ { - 1 } \right)\).

6 The continuous random variable \(X\) has (cumulative) distribution function given by $$\mathrm { F } ( x ) = \begin{cases} 0 & x < \frac { 1 } { 2 } \\ \frac { 2 x - 1 } { x + 1 } & \frac { 1 } { 2 } \leqslant x \leqslant 2 \\ 1 & x > 2 . \end{cases}$$

1. Given that \(Y = \frac { 1 } { X }\), find the (cumulative) distribution function of \(Y\), and deduce that \(Y\) and \(X\) have identical distributions.
2. Find \(\mathrm { E } ( X + 1 )\) and deduce the value of \(\mathrm { E } \left( \frac { 1 } { X } \right)\).

1 Design engineers are simulating the load on a particular part of a complex structure. They intend that the simulated load, measured in a convenient unit, should be given by the random variable \(X\) having probability density function $$f ( x ) = 12 x ^ { 3 } - 24 x ^ { 2 } + 12 x , \quad 0 \leqslant x \leqslant 1 .$$

1. Find the mean and the mode of \(X\).
2. Find the cumulative distribution function \(\mathrm { F } ( x )\) of \(X\). $$\text { Verify that } \mathrm { F } \left( \frac { 1 } { 4 } \right) = \frac { 67 } { 256 } , \mathrm {~F} \left( \frac { 1 } { 2 } \right) = \frac { 11 } { 16 } \text { and } \mathrm { F } \left( \frac { 3 } { 4 } \right) = \frac { 243 } { 256 } .$$ The engineers suspect that the process for generating simulated loads might not be working as intended. To investigate this, they generate a random sample of 512 loads. These are recorded in a frequency distribution as follows.

   Load \(x\)	\(0 \leqslant x \leqslant \frac { 1 } { 4 }\)	\(\frac { 1 } { 4 } < x \leqslant \frac { 1 } { 2 }\)	\(\frac { 1 } { 2 } < x \leqslant \frac { 3 } { 4 }\)	\(\frac { 3 } { 4 } < x \leqslant 1\)
   Frequency	126	209	131	46

3. Use a suitable statistical procedure to assess the goodness of fit of \(X\) to these data. Discuss your conclusions briefly.

9 A rectangle of area \(A \mathrm {~m} ^ { 2 }\) has a perimeter of 20 m and each of the two shorter sides are of length \(X \mathrm {~m}\), where \(X\) is uniformly distributed between 0 and 2 .

1. Write down an expression for \(A\) in terms of \(X\), and hence show that \(A = 25 - ( X - 5 ) ^ { 2 }\).
2. Write down the probability density function of \(X\).
3. Show that the cumulative distribution function of \(A\) is $$\mathrm { F } ( a ) = \left\{ \begin{array} { l r } 0 & a < 0 , \\ \frac { 1 } { 2 } ( 5 - \sqrt { 25 - a } ) & 0 \leqslant a \leqslant 16 , \\ 1 & a > 16 . \end{array} \right.$$
4. Find the probability density function of \(A\). \section*{END OF QUESTION PAPER} \section*{OCR}

7 A continuous random variable \(X\) has probability density function $$f ( x ) = \left\{ \begin{array} { c c } k x & 0 \leqslant x < 2 \\ \frac { k ( 4 - x ) ^ { 2 } } { 2 } & 2 \leqslant x \leqslant 4 \\ 0 & \text { otherwise } \end{array} \right.$$ where \(k\) is a constant.

1. Show that \(k = \frac { 3 } { 10 }\).
2. Find \(\mathrm { E } ( X )\).
3. Find the cumulative distribution function of \(X\).
4. Find the upper quartile of \(X\), correct to 3 significant figures. \section*{END OF QUESTION PAPER}

2 The continuous random variable \(U\) has (cumulative) distribution function given by $$\mathrm { F } ( u ) = \begin{cases} \frac { 1 } { 5 } \mathrm { e } ^ { u } & u < 0 \\ 1 - \frac { 4 } { 5 } \mathrm { e } ^ { - \frac { 1 } { 4 } u } & u \geqslant 0 \end{cases}$$

1. Find the upper quartile of \(U\).
2. Find the probability density function of \(U\).

6 \includegraphics[max width=\textwidth, alt={}, center]{054e0081-afce-4a87-93f5-650dad40b313-3_508_611_262_719} The diagram shows the probability density function f of the continuous random variable \(T\), given by $$f ( t ) = \begin{cases} a t & 0 \leqslant t \leqslant 1 \\ a & 1 < t \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$ where \(a\) is a constant.

1. Find the value of \(a\).
2. Obtain the cumulative distribution function of \(T\).
3. Find the cumulative distribution of \(Y\), where \(Y = T ^ { \frac { 1 } { 2 } }\), and hence find the probability density function of \(Y\).

5 The continuous random variable \(Y\) has probability density function given by $$\mathrm { f } ( y ) = \begin{cases} \ln ( y ) & 1 \leqslant y \leqslant \mathrm { e } \\ 0 & \text { otherwise } \end{cases}$$

1. Verify that this is a valid probability density function.
2. Show that the (cumulative) distribution function of \(Y\) is given by $$\mathrm { F } ( y ) = \begin{cases} 0 & y < 1 \\ y \ln y - y + 1 & 1 \leqslant y \leqslant \mathrm { e } \\ 1 & \text { otherwise } \end{cases}$$
3. Verify that the upper quartile of \(Y\) lies in the interval [2.45, 2.46].
4. Find the (cumulative) distribution function of \(X\) where \(X = \ln Y\).

2

1. A continuous random variable, \(X\), has probability density function $$f ( x ) = \begin{cases} \frac { 1 } { 72 } \left( 8 x - x ^ { 2 } \right) & 2 \leqslant x \leqslant 8 \\ 0 & \text { otherwise } \end{cases}$$
   1. Find \(\mathrm { F } ( x )\), the cumulative distribution function of \(X\).
   2. Sketch \(\mathrm { F } ( x )\).
   3. The median of \(X\) is \(m\). Show that \(m\) satisfies the equation \(m ^ { 3 } - 12 m ^ { 2 } + 148 = 0\). Verify that \(m \approx 4.42\).
2. The random variable in part (a) is thought to model the weights, in kilograms, of lambs at birth. The birth weights, in kilograms, of a random sample of 12 lambs, given in ascending order, are as follows. $$\begin{array} { l l l l l l l l l l l l } 3.16 & 3.62 & 3.80 & 3.90 & 4.02 & 4.72 & 5.14 & 6.36 & 6.50 & 6.58 & 6.68 & 6.78 \end{array}$$ Test at the 5\% level of significance whether a median of 4.42 is consistent with these data.

3 The time, in hours, until an electronic component fails is represented by the random variable \(X\). In this question two models for \(X\) are proposed.

1. In one model, \(X\) has cumulative distribution function $$\mathrm { G } ( x ) = \begin{cases} 0 & x \leqslant 0 \\ 1 - \left( 1 + \frac { x } { 200 } \right) ^ { - 2 } & x > 0 \end{cases}$$ (A) Sketch \(\mathrm { G } ( x )\).
   (B) Find the interquartile range for this model. Hence show that a lifetime of more than 454 hours (to the nearest hour) would be classed as an outlier.
2. In the alternative model, \(X\) has probability density function $$\mathrm { f } ( x ) = \begin{cases} \frac { 1 } { 200 } \mathrm { e } ^ { - \frac { 1 } { 200 } x } & x > 0 \\ 0 & \text { elsewhere. } \end{cases}$$ (A) For this model show that the cumulative distribution function of \(X\) is $$\mathrm { F } ( x ) = \begin{cases} 0 & x \leqslant 0 \\ 1 - \mathrm { e } ^ { - \frac { 1 } { 200 } x } & x > 0 \end{cases}$$ (B) Show that \(\mathrm { P } ( X > 50 ) = \mathrm { e } ^ { - 0.25 }\).
   (C) It is observed that a particular component is still working after 400 hours. Find the conditional probability that it will still be working after a further 50 hours (i.e. after a total of 450 hours) given that it is still working after 400 hours.

3 The random variable \(X\) has the following probability density function, \(\mathrm { f } ( x )\). $$f ( x ) = \begin{cases} k x ( x - 5 ) ^ { 2 } & 0 \leqslant x < 5 \\ 0 & \text { elsewhere } \end{cases}$$

1. Sketch \(\mathrm { f } ( x )\).
2. Find, in terms of \(k\), the cumulative distribution function, \(\mathrm { F } ( x )\).
3. Hence show that \(k = \frac { 12 } { 625 }\). The random variable \(X\) is proposed as a model for the amount of time, in minutes, lost due to stoppages during a football match. The times lost in a random sample of 60 matches are summarised in the table. The table also shows some of the corresponding expected frequencies given by the model.

   Time (minutes)	\(0 \leqslant x < 1\)	\(1 \leqslant x < 2\)	\(2 \leqslant x < 3\)	\(3 \leqslant x < 4\)	\(4 \leqslant x < 5\)
   Observed frequency	5	15	23	11	6
   Expected frequency			17.76	9.12	1.632

4. Find the remaining expected frequencies.
5. Carry out a goodness of fit test, using a significance level of \(2.5 \%\), to see if the model might be suitable in this context.

The continuous random variable \(T\) has probability density function given by $$\mathrm { f } ( t ) = \begin{cases} 0 & t < 2 \\ \frac { 2 } { ( t - 1 ) ^ { 3 } } & t \geqslant 2 \end{cases}$$

1. Find the distribution function of \(T\), and find also \(\mathrm { P } ( T > 5 )\).
2. Consecutive independent observations of \(T\) are made until the first observation that exceeds 5 is obtained. The random variable \(N\) is the total number of observations that have been made up to and including the observation exceeding 5. Find \(\mathrm { P } ( N > \mathrm { E } ( N ) )\).
3. Find the probability density function of \(Y\), where \(Y = \frac { 1 } { T - 1 }\).

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 \(Y\).

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 \(Y\).

9 The continuous random variable \(X\) has probability density function f given by $$f ( x ) = \begin{cases} \frac { 1 } { 2 a } & - a \leqslant x \leqslant a \\ 0 & \text { otherwise } \end{cases}$$ where \(a\) is a positive constant. Find the distribution function of \(X\). The random variable \(Y\) is defined by \(Y = \mathrm { e } ^ { X }\). Find the distribution function of \(Y\). Given that \(a = 4\), find the value of \(k\) for which \(\mathrm { P } ( Y \geqslant k ) = 0.25\).

6 The random variable \(X\) has distribution function F given by $$\mathrm { F } ( x ) = \begin{cases} 1 - \mathrm { e } ^ { - 0.6 x } & x \geqslant 0 \\ 0 & \text { otherwise } \end{cases}$$ Identify the distribution of \(X\) and state its mean. Find

1. \(\mathrm { P } ( X > 4 )\),
2. the median of \(X\).

9 The continuous random variable \(X\) has probability density function given by $$\mathrm { f } ( x ) = \begin{cases} 0 & x < 2 \\ a \mathrm { e } ^ { - ( x - 2 ) } & x \geqslant 2 \end{cases}$$ where \(a\) is a constant. Show that \(a = 1\). Find the distribution function of \(X\) and hence find the median value of \(X\). The random variable \(Y\) is defined by \(Y = \mathrm { e } ^ { X }\). Find

1. the probability density function of \(Y\),
2. \(\mathrm { P } ( Y > 10 )\).

8 The random variable \(X\) has probability density function f given by $$\mathrm { f } ( x ) = \begin{cases} 2 \mathrm { e } ^ { - 2 x } & x \geqslant 0 \\ 0 & \text { otherwise } \end{cases}$$

1. Find the distribution function of \(X\).
2. Find the median value of \(X\). The random variable \(Y\) is defined by \(Y = \mathrm { e } ^ { X }\).
3. Find the probability density function of \(Y\).

6 The continuous random variable \(X\) has distribution function given by $$\mathrm { F } ( x ) = \begin{cases} 1 - \mathrm { e } ^ { - 0.4 x } & x \geqslant 0 \\ 0 & \text { otherwise } \end{cases}$$

1. Find \(\mathrm { P } ( X > 2 )\).
2. Find the interquartile range of \(X\).

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

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