Find or specify CDF

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.

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

3 For a restaurant with a home-delivery service, the delivery time in minutes can be modelled by a continuous random variable \(T\) with probability density function given by $$f ( t ) = \begin{cases} \frac { \pi } { 90 } \sin \left( \frac { \pi t } { 60 } \right) & 20 \leqslant t \leqslant 60
0 & \text { otherwise. } \end{cases}$$

1. Given that \(20 \leqslant a \leqslant 60\), show that \(\mathrm { P } ( T \leqslant a ) = \frac { 1 } { 3 } \left( 1 - 2 \cos \left( \frac { \pi a } { 60 } \right) \right)\). There is a delivery charge of \(\pounds 3\) but this is reduced to \(\pounds 2\) if the delivery time exceeds a minutes.
2. Find the value of \(a\) for which the expected value of the delivery charge for a home-delivery is £2.80.

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 \(\mathrm { E } ( \sqrt { Y } )\).

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 \(Y\).

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 \(Y = \frac { 1 } { T - 1 }\).

4 The continuous random variable \(X\) has a probability density function given by $$\mathrm { f } ( x ) = \begin{cases} \frac { 1 } { 2 } k ( x - 1 ) & 1 \leqslant x \leqslant 3
k & 3 < x \leqslant 6
\frac { 1 } { 4 } k ( 10 - x ) & 6 < x \leqslant 10
0 & \text { otherwise } \end{cases}$$ where \(k\) is a positive constant.

1. Sketch \(\mathrm { f } ( x )\) for all values of \(x\)
2. Show that \(k = \frac { 1 } { 6 }\)
3. Specify fully the cumulative distribution function \(\mathrm { F } ( x )\) of \(X\) Given that \(\mathrm { E } ( X ) = \frac { 61 } { 12 }\)
4. find \(\mathrm { P } ( X > \mathrm { E } ( X ) )\)
5. Describe the skewness of the distribution, giving a reason for your answer.
   

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

$$f ( x ) = \begin{cases} k \left( x ^ { 2 } - 2 x + 2 \right) & 0 < x \leqslant 3
3 k & 3 < x \leqslant 4
0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = \frac { 1 } { 9 }\)
2. Find the cumulative distribution function \(\mathrm { F } ( x )\).
3. Find the mean of \(X\).
4. Show that the median of \(X\) lies between \(x = 2.6\) and \(x = 2.7\)

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

1. Sketch the graph of \(\mathrm { f } ( x )\).
2. Show that \(k = \frac { 1 } { 2 } ( 1 + \sqrt { 5 } )\).
3. Define fully the cumulative distribution function \(\mathrm { F } ( x )\).
4. Find \(\mathrm { P } ( 0.5 < X < 1.5 )\).
5. Write down the median of \(X\) and the mode of \(X\).
6. Describe the skewness of the distribution of \(X\). Give a reason for your answer.

7 Engineering work on the railway network causes an increase in the journey time of commuters travelling into work each morning. The increase in journey time, \(T\) hours, is modelled by a continuous random variable with probability density function $$\mathrm { f } ( t ) = \begin{cases} 4 t \left( 1 - t ^ { 2 } \right) & 0 \leqslant t \leqslant 1
0 & \text { otherwise } \end{cases}$$

1. Show that \(\mathrm { E } ( T ) = \frac { 8 } { 15 }\).
2. 1. Find the cumulative distribution function, \(\mathrm { F } ( t )\), for \(0 \leqslant t \leqslant 1\).
   2. Hence, or otherwise, for a commuter selected at random, find $$\mathrm { P } ( \text { mean } < T < \text { median } )$$

6 The waiting time, \(T\) minutes, before being served at a local newsagents can be modelled by a continuous random variable with probability density function $$\mathrm { f } ( t ) = \begin{cases} \frac { 3 } { 8 } t ^ { 2 } & 0 \leqslant t < 1
\frac { 1 } { 16 } ( t + 5 ) & 1 \leqslant t \leqslant 3
0 & \text { otherwise } \end{cases}$$

1. Sketch the graph of f.
2. For a customer selected at random, calculate \(\mathrm { P } ( T \geqslant 1 )\).
3. 1. Show that the cumulative distribution function for \(1 \leqslant t \leqslant 3\) is given by $$\mathrm { F } ( t ) = \frac { 1 } { 32 } \left( t ^ { 2 } + 10 t - 7 \right)$$
   2. Hence find the median waiting time.

7. A continuous random variable \(X\) has a probability density function given by $$\begin{array} { l l } \mathrm { f } ( x ) = \frac { x ^ { 2 } } { 312 } & 4 \leq x \leq 10
\mathrm { f } ( x ) = 0 & \text { otherwise. } \end{array}$$

1. Find \(\mathrm { E } ( X )\).
2. Find the variance of \(X\).
3. Find the cumulative distribution function \(\mathrm { F } ( x )\), for all values of \(x\).
4. Hence find the median value of \(X\).
5. Write down the modal value of \(X\). It is sometimes suggested that, for most distributions, $$2 \times ( \text { median } - \text { mean } ) \approx \text { mode } - \text { median } .$$
6. Show that this result is not satisfied in this case, and suggest a reason why.

4. The continuous random variable \(R\) has probability density function \(f ( r )\) given by $$f ( r ) = \begin{cases} k r ( b - r ) & \text { for } 1 \leqslant r \leqslant 4 ,
0 & \text { otherwise } , \end{cases}$$ where \(k\) and \(b\) are positive constants.

1. Explain why \(b \geqslant 4\).
2. Given that \(b = 4\),
   1. show that \(k = \frac { 1 } { 9 }\),
   2. find an expression for \(F ( r )\), valid for \(1 \leqslant r \leqslant 4\), where \(F\) denotes the cumulative distribution function of \(R\),
   3. find the probability that \(R\) lies between 2 and 3 .

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.

1. A continuous random variable \(X\) has probability density function

$$f ( x ) = \left\{ \begin{array} { c c } \frac { x } { 16 } \left( 9 - x ^ { 2 } \right) & 1 \leqslant x \leqslant 3
0 & \text { otherwise } \end{array} \right.$$

1. Find the cumulative distribution function of \(X\)
2. Calculate \(\mathrm { P } ( X > 1.8 )\)
3. Use calculus to find \(\mathrm { E } \left( \frac { 3 } { X } + 2 \right)\)
4. Show that the mode of \(X\) is \(\sqrt { 3 }\)

7. The continuous random variable \(X\) has probability density function given by $$f ( x ) = \left\{ \begin{array} { c l } 0 & x < 1
\frac { 4 } { x ^ { 5 } } & x \geq 1 \end{array} \right.$$ a. Find the cumulative distribution function, \(F ( x )\), of \(X\).
b. Find the interquartile range of \(X\).
c. Show that the probability density function of \(Y\), where \(Y = \frac { 1 } { X ^ { 2 } }\), is given by $$g ( y ) = \left\{ \begin{array} { c l } 2 y & 0 < y \leq 1
0 & \text { otherwise } \end{array} \right.$$ d. Find the value of \(a\) for which \(\mathrm { E } \left( \frac { 1 } { X ^ { 2 } } \right) = a \mathrm { E } \left( X ^ { 2 } \right)\).

7 The continuous random variable \(X\) has the probability density function given by $$f ( x ) = \left\{ \begin{array} { c c } \frac { 1 } { 16 } x ^ { 3 } & 0 \leqslant x \leqslant 2
\frac { 1 } { 6 } ( 5 - x ) & 2 \leqslant x \leqslant 5
0 & \text { otherwise } \end{array} \right.$$

1. Sketch the graph of f.
2. Prove that the cumulative distribution function of \(X\) for \(2 \leqslant x \leqslant 5\) can be written in the form $$\mathrm { F } ( x ) = 1 - \frac { 1 } { 12 } ( 5 - x ) ^ { 2 }$$
3. Hence, or otherwise, determine \(\mathrm { P } ( X \geqslant 3 \mid X \leqslant 4 )\).

8 The continuous random variable \(X\) has probability density function $$f ( x ) = \begin{cases} k \sin 2 x & 0 \leq x \leq \frac { \pi } { 6 }
0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant. 8

1. Show that \(k = 4\)
   8
2. Find the cumulative distribution function \(\mathrm { F } ( x )\)
   8
3. Find the median of \(X\), giving your answer to three significant figures. 8
4. Find the mean of \(X\) giving your answer in the form \(\frac { 1 } { a } ( b \sqrt { 3 } - \pi )\) where \(a\) and \(b\) are integers.
   \includegraphics[max width=\textwidth, alt={}, center]{1e2fdd33-afa4-486f-a9e2-1d425ed14eee-14_2492_1721_217_150}