5.03e Find cdf: by integration

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.

8 The continuous random variable \(X\) has the cumulative distribution function $$\mathrm { F } ( x ) = \left\{ \begin{array} { c c } 0 & x \leqslant - 4 \\ \frac { x + 4 } { 9 } & - 4 \leqslant x \leqslant 5 \\ 1 & x \geqslant 5 \end{array} \right.$$

1. Determine the probability density function, \(\mathrm { f } ( x )\), of \(X\).
2. Sketch the graph of f .
3. Determine \(\mathrm { P } ( X > 2 )\).
4. Evaluate the mean and variance of \(X\).

7 The waiting time, \(X\) minutes, for fans to gain entrance to see an event may be modelled by a continuous random variable having the distribution function defined by $$\mathrm { F } ( x ) = \begin{cases} 0 & x < 0 \\ \frac { 1 } { 2 } x & 0 \leqslant x \leqslant 1 \\ \frac { 1 } { 54 } \left( x ^ { 3 } - 12 x ^ { 2 } + 48 x - 10 \right) & 1 \leqslant x \leqslant 4 \\ 1 & x > 4 \end{cases}$$

1. 1. Sketch the graph of F.
   2. Explain why the value of \(q _ { 1 }\), the lower quartile of \(X\), is \(\frac { 1 } { 2 }\).
   3. Show that the upper quartile, \(q _ { 3 }\), satisfies \(1.6 < q _ { 3 } < 1.7\).
2. The probability density function of \(X\) is defined by $$\mathrm { f } ( x ) = \begin{cases} \alpha & 0 \leqslant x \leqslant 1 \\ \beta ( x - 4 ) ^ { 2 } & 1 \leqslant x \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$
   1. Show that the exact values of \(\alpha\) and \(\beta\) are \(\frac { 1 } { 2 }\) and \(\frac { 1 } { 18 }\) respectively.
   2. Hence calculate \(\mathrm { E } ( X )\).

6 The random variable \(X\) has probability density function defined by $$f ( x ) = \begin{cases} \frac { 1 } { 40 } ( x + 7 ) & 1 \leqslant x \leqslant 5 \\ 0 & \text { otherwise } \end{cases}$$

1. Sketch the graph of f.
2. Find the exact value of \(\mathrm { E } ( X )\).
3. Prove that the distribution function F , for \(1 \leqslant x \leqslant 5\), is defined by $$\mathrm { F } ( x ) = \frac { 1 } { 80 } ( x + 15 ) ( x - 1 )$$
4. Hence, or otherwise:
   1. find \(\mathrm { P } ( 2.5 \leqslant X \leqslant 4.5 )\);
   2. show that the median, \(m\), of \(X\) satisfies the equation \(m ^ { 2 } + 14 m - 55 = 0\).
5. Calculate the value of the median of \(X\), giving your answer to three decimal places.

6 The time, in weeks, that a patient must wait to be given an appointment in Holmsoon Hospital may be modelled by a random variable \(T\) having the cumulative distribution function $$\mathrm { F } ( t ) = \begin{cases} 0 & t < 0 \\ \frac { t ^ { 3 } } { 216 } & 0 \leqslant t \leqslant 6 \\ 1 & t > 6 \end{cases}$$

1. Find, to the nearest day, the time within which 90 per cent of patients will have been given an appointment.
2. Find the probability density function of \(T\) for all values of \(t\).
3. Calculate the mean and the variance of \(T\).
4. Calculate the probability that the time that a patient must wait to be given an appointment is more than one standard deviation above the mean.

7 The continuous random variable \(X\) has probability density function defined by $$f ( x ) = \begin{cases} \frac { 1 } { 5 } ( 2 x + 1 ) & 0 \leqslant x \leqslant 1 \\ \frac { 1 } { 15 } ( 4 - x ) ^ { 2 } & 1 < x \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$

1. Sketch the graph of f.
2. 1. Show that the cumulative distribution function, \(\mathrm { F } ( x )\), for \(0 \leqslant x \leqslant 1\) is $$\mathrm { F } ( x ) = \frac { 1 } { 5 } x ( x + 1 )$$
   2. Hence write down the value of \(\mathrm { P } ( X \leqslant 1 )\).
   3. Find the value of \(x\) for which \(\mathrm { P } ( X \geqslant x ) = \frac { 17 } { 20 }\).
   4. Find the lower quartile of the distribution.

8 The continuous random variable \(X\) has cumulative distribution function $$\mathrm { F } ( x ) = \left\{ \begin{array} { c c } 0 & x < - 1 \\ \frac { x + 1 } { k + 1 } & - 1 \leqslant x \leqslant k \\ 1 & x > k \end{array} \right.$$ where \(k\) is a positive constant.

1. Find, in terms of \(k\), an expression for \(\mathrm { P } ( X < 0 )\).
2. Determine an expression, in terms of \(k\), for the lower quartile, \(q _ { 1 }\).
3. Show that the probability density function of \(X\) is defined by $$\mathrm { f } ( x ) = \left\{ \begin{array} { c c } \frac { 1 } { k + 1 } & - 1 \leqslant x \leqslant k \\ 0 & \text { otherwise } \end{array} \right.$$
4. Given that \(k = 11\) :
   1. sketch the graph of f;
   2. determine \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\);
   3. show that \(\mathrm { P } \left( q _ { 1 } < X < \mathrm { E } ( X ) \right) = 0.25\).

6 The continuous random variable \(X\) has the probability density function defined by $$f ( x ) = \begin{cases} \frac { 3 } { 8 } \left( x ^ { 2 } + 1 \right) & 0 \leqslant x \leqslant 1 \\ \frac { 1 } { 4 } ( 5 - 2 x ) & 1 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{cases}$$

1. The cumulative distribution function of \(X\) is denoted by \(\mathrm { F } ( x )\). Show that, for \(0 \leqslant x \leqslant 1\), $$\mathrm { F } ( x ) = \frac { 1 } { 8 } x \left( x ^ { 2 } + 3 \right)$$
2. Hence, or otherwise, verify that the median value of \(X\) is 1 .
3. Show that the upper quartile, \(q\), satisfies the equation \(q ^ { 2 } - 5 q + 5 = 0\) and hence that \(q = \frac { 1 } { 2 } ( 5 - \sqrt { 5 } )\).
4. Calculate the exact value of \(\mathrm { P } ( q < X < 1.5 )\).

3 The continuous random variable \(X\) has a cumulative distribution function defined by $$\mathrm { F } ( x ) = \left\{ \begin{array} { c l } 0 & x < - 5 \\ \frac { x + 5 } { 20 } & - 5 \leqslant x \leqslant 15 \\ 1 & x > 15 \end{array} \right.$$

1. Show that, for \(- 5 \leqslant x \leqslant 15\), the probability density function, \(\mathrm { f } ( x )\), of \(X\) is given by \(\mathrm { f } ( x ) = \frac { 1 } { 20 }\).
   (1 mark)
2. Find:
   1. \(\mathrm { P } ( X \geqslant 7 )\);
   2. \(\mathrm { P } ( X \neq 7 )\);
   3. \(\mathrm { E } ( X )\);
   4. \(\mathrm { E } \left( 3 X ^ { 2 } \right)\).

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

1. Sketch the graph of f on the axes below.
2. 1. Find the cumulative distribution function, F , for \(0 \leqslant x \leqslant 1\).
   2. Hence, or otherwise, find the value of the lower quartile of \(X\).
3. 1. Show that the cumulative distribution function for \(1 \leqslant x \leqslant 2\) is defined by $$\mathrm { F } ( x ) = \frac { 1 } { 3 } \left( 5 x - x ^ { 2 } - 3 \right)$$
   2. Hence, or otherwise, find the value of the upper quartile of \(X\). \includegraphics[max width=\textwidth, alt={}, center]{03c1e107-3377-4b0d-9daf-7f70233c18b5-5_554_1050_1217_424}

6 The continuous random variable \(X\) has the cumulative distribution function $$\mathrm { F } ( x ) = \begin{cases} 0 & x < 0 \\ \frac { 1 } { 2 } x - \frac { 1 } { 16 } x ^ { 2 } & 0 \leqslant x \leqslant 4 \\ 1 & x > 4 \end{cases}$$

1. Find the probability that \(X\) lies between 0.4 and 0.8 .
2. Show that the probability density function, \(\mathrm { f } ( x )\), of \(X\) is given by $$f ( x ) = \begin{cases} \frac { 1 } { 2 } - \frac { 1 } { 8 } x & 0 \leqslant x \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$
3. 1. Find the value of \(\mathrm { E } ( X )\).
   2. Show that \(\operatorname { Var } ( X ) = \frac { 8 } { 9 }\).
4. The continuous random variable \(Y\) is defined by $$Y = 3 X - 2$$ Find the values of \(\mathrm { E } ( Y )\) and \(\operatorname { Var } ( Y )\).

4. A continuous random variable \(X\) has the cumulative distribution function $$\mathrm { F } ( x ) = \left\{ \begin{array} { l } 0 \\ \frac { 1 } { 84 } \left( x ^ { 2 } - 16 \right) \\ 1 \end{array} \right.$$ $$\begin{aligned} & x < 4 , \\ & 4 \leq x \leq 10 , \\ & x > 10 . \end{aligned}$$

1. Find the median value of \(X\).
2. Find the interquartile range for \(X\).
3. Find the probability density function \(\mathrm { f } ( x )\) of \(X\).
4. Sketch the graph of \(\mathrm { f } ( x )\) and hence write down the mode of \(X\), explaining how you obtain your answer from the graph. \section*{STATISTICS 2 (A) TEST PAPER 1 Page 2}

2. A continuous random variable \(X\) has the probability density function $$\begin{array} { l l } \mathrm { f } ( x ) = k & 5 \leq x \leq 15 , \\ \mathrm { f } ( x ) = 0 & \text { otherwise. } \end{array}$$

1. Find \(k\) and specify the cumulative density function \(\mathrm { F } ( x )\).
2. Write down the value of \(\mathrm { P } ( X < 8 )\).

7. Each day on the way to work, a commuter encounters a similar traffic jam. The length of time, in 10-minute units, spent waiting in the traffic jam is modelled by the random variable \(T\) with the cumulative distribution function: $$\begin{array} { l l } \mathrm { F } ( t ) = 0 & t < 0 , \\ \mathrm {~F} ( t ) = \frac { t ^ { 2 } \left( 3 t ^ { 2 } - 16 t + 24 \right) } { 16 } & 0 \leq t \leq 2 , \\ \mathrm {~F} ( t ) = 1 & t > 2 . \end{array}$$

1. Show that 0.77 is approximately the median value of \(T\).
2. Given that he has already waited for 12 minutes, find the probability that he will have to wait another 3 minutes.
3. Find, and sketch, the probability density function of \(T\).
4. Hence find the modal value of \(T\).
5. Comment on the validity of this model.

1. (a) Explain the difference between a discrete and a continuous variable.

A random number generator on a calculator generates numbers, \(X\), to 3 decimal places, in the range 0 to 1 , e.g. 0.386 . The variable \(X\) may be modelled by a continuous uniform distribution, having the probability density function \(\mathrm { f } ( x )\), where $$\begin{array} { l l } \mathrm { f } ( x ) = 1 & 0 < x < 1 \\ \mathrm { f } ( x ) = 0 & \text { otherwise } \end{array}$$ (b) Explain why this model is not totally accurate.
(c) Sketch the cumulative distribution function of \(X\).

7. The fraction of sky covered by cloud is modelled by the random variable \(X\) with probability density function $$\begin{array} { l l } \mathrm { f } ( x ) = 0 & x < 0 \\ \mathrm { f } ( x ) = k x ^ { 2 } ( 1 - x ) & 0 \leq x \leq 1 , \\ \mathrm { f } ( x ) = 0 & x > 1 . \end{array}$$

1. Find \(k\) and sketch the graph of \(\mathrm { f } ( x )\).
2. Find the mean and the variance of \(X\).
3. Find the cumulative distribution function \(\mathrm { F } ( x )\).
4. Given that flying is prohibited when \(85 \%\) of the sky is covered by cloud, show that cloud conditions allow flying nearly \(90 \%\) of the time.

1. The continuous random variable \(X\) has the following cumulative distribution function:

$$F ( x ) = \begin{cases} 0 , & x < 2 \\ k \left( 19 x - x ^ { 2 } - 34 \right) , & 2 \leq x \leq 5 \\ 1 , & x > 5 \end{cases}$$

1. Show that \(k = \frac { 1 } { 36 }\).
2. Find \(\mathrm { P } ( X > 4 )\).
3. Find and specify fully the probability density function \(\mathrm { f } ( x )\) of \(X\).

6. The continuous random variable \(X\) has the following probability density function: $$f ( x ) = \begin{cases} \frac { 1 } { 16 } x , & 2 \leq x \leq 6 \\ 0 , & \text { otherwise } \end{cases}$$

1. Sketch \(\mathrm { f } ( x )\) for all values of \(x\).
2. Find \(\mathrm { E } ( X )\).
3. Show that \(\operatorname { Var } ( X ) = \frac { 11 } { 9 }\).
4. Define fully the cumulative distribution function \(\mathrm { F } ( x )\) of \(X\).
5. Show that the interquartile range of \(X\) is \(2 ( \sqrt { } 7 - \sqrt { 3 } )\). END

4 The cumulative distribution function of the continuous random variable \(X\) is given by \(\mathrm { F } ( x ) = \begin{cases} 0 & x < 0 , \\ k \left( 12 x - x ^ { 2 } \right) & 0 \leqslant x \leqslant 2 , \\ 1 & x > 2 , \end{cases}\) where \(k\) is a constant.

1. Show that \(k = 0.05\).
2. Find \(\mathrm { P } ( 1 \leqslant X \leqslant 1.5 )\).
3. Find the median of \(X\), correct to 3 significant figures.
4. Find which of the median, mean and mode of \(X\) is the largest of the three measures of central tendency.

2 The cumulative distribution function of the continuous random variable, \(Y\), is given below. $$\mathrm { F } ( y ) = \left\{ \begin{array} { c c } 0 & y < 0 \\ \frac { y ^ { 3 } - y ^ { 2 } } { 4 } & 1 \leq y \leq 2 \\ 1 & y > 2 \end{array} \right.$$

1. Find \(\mathrm { P } ( Y \leq 1.5 )\)
2. Verify that the median of \(Y\) lies between 1.6 and 1.7.
3. Find the probability density function of \(Y\).

10 The probability density function of the continuous random variable \(X\) is given by \(f ( x ) = \begin{cases} k x ^ { m } & 0 \leqslant x \leqslant a , \\ 0 & \text { otherwise, } \end{cases}\) where \(a , k\) and \(m\) are positive constants.

1. Show that \(k = \frac { m + 1 } { a ^ { m + 1 } }\).
2. Find the cumulative distribution function of \(X\) in terms of \(x , a\) and \(m\).
3. Given that \(\mathrm { P } \left( \frac { 1 } { 4 } a < X < \frac { 1 } { 2 } a \right) = \frac { 1 } { 10 }\),
   1. show that \(2 p ^ { 2 } - 10 p + 5 = 0\), where \(p = 2 ^ { m }\),
   2. find the value of \(m\). \section*{END OF QUESTION PAPER}

10 The continuous random variable \(X\) has probability density function given by \(f ( x ) = \begin{cases} \frac { 4 } { 15 } \left( \frac { a } { x ^ { 2 } } + 3 x ^ { 2 } - \frac { 7 } { 2 } \right) & 1 \leqslant x \leqslant 2 , \\ 0 & \text { otherwise, } \end{cases}\) where \(a\) is a positive constant.

1. Find the cumulative distribution function of \(X\) in terms of \(a\).
2. Hence or otherwise determine the value of \(a\).
3. Show that the median value \(m\) of \(X\) satisfies the equation $$8 m ^ { 4 } - 28 m ^ { 2 } + 9 m - 4 = 0 .$$
4. Verify that the median value of \(X\) is 1.74, correct to \(\mathbf { 2 }\) decimal places.
5. Find \(\mathrm { E } ( X )\).
6. Determine the mode of \(X\).

12 The cumulative distribution function of the continuous random variable \(X\) is given by \(F ( x ) = \begin{cases} 0 & x < 20 , \\ a \left( x ^ { 2 } + b x + c \right) & 20 \leqslant x \leqslant 30 , \\ 1 & x > 30 , \end{cases}\) where \(a\), \(b\) and \(c\) are constants.
You are given that \(\mathrm { P } ( X < 25 ) = \frac { 11 } { 24 }\).

1. Find \(\mathrm { P } ( X > 27 )\).
2. Find the 90th percentile of \(X\).

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).

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 .