5.03b Solve problems: using pdf

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.

2. The random variable \(X\), which can take any value in the interval \(1 \leq X \leq n\), is modelled by the continuous uniform distribution with mean 12.

1. Show that \(n = 23\) and find the variance of \(X\).
2. Find \(\mathrm { P } ( 10 < X < 14 )\).

4. A continuous random variable \(X\) has probability density function $$\begin{array} { l l } \mathrm { f } ( x ) = 0 & x < 1 , \\ \mathrm { f } ( x ) = k x & 1 \leq x \leq 4 , \\ \mathrm { f } ( x ) = 0 & x > 4 . \end{array}$$

1. Sketch a graph of \(\mathrm { f } ( x )\), and hence find the value of \(k\).
2. Calculate the mean and the variance of \(X\). \section*{STATISTICS 2 (A)TEST PAPER 4 Page 2}

7. The time, in hours, taken to run the London marathon is modelled by a continuous random variable \(T\) with the probability density function $$f ( t ) = \begin{cases} c ( t - 2 ) & 2 \leq t < 4 \\ \frac { 2 c ( 7 - t ) } { 3 } & 4 \leq t \leq 7 \\ 0 & \text { otherwise } \end{cases}$$

1. Sketch the function \(\mathrm { f } ( t )\), and show that \(c = \frac { 1 } { 5 }\).
2. Calculate the median value of \(T\).
3. Make two critical comments about the model.

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 \(X\).
   

5. In a party game, a bottle is spun and whoever it points to when it stops has to play next. The acute angle, in degrees, that the bottle makes with the side of the room is modelled by a rectangular distribution over the interval [0,90]. Find the probability that on one spin this angle is

1. between \(25 ^ { \circ }\) and \(38 ^ { \circ }\),
2. \(45 ^ { \circ }\) to the nearest degree. The bottle is spun ten times.
3. Find the probability that the acute angle it makes with the side of the room is less than \(10 ^ { \circ }\) more than twice.

7. In a competition at a funfair, participants have to stay on a log being rotated in a pool of water for as long as possible. The length of time, in tens of seconds, that the competitors stay on the log is modelled by the random variable \(T\) with the following probability density function: $$\mathrm { f } ( t ) = \begin{cases} k ( t - 3 ) ^ { 2 } , & 0 \leq t \leq 3 \\ 0 , & \text { otherwise } \end{cases}$$

1. Show that \(k = \frac { 1 } { 9 }\).
2. Sketch f \(( t )\) for all values of \(t\).
3. Show that the mean time that competitors stay on the \(\log\) is 7.5 seconds. When the competition is next run the organisers decide to make it easier at first by spinning the log more slowly and then increasing the speed of rotation. The length of time, in tens of seconds, that the competitors now stay on the log is modelled by the random variable \(S\) with the following probability density function: $$f ( s ) = \begin{cases} \frac { 1 } { 12 } \left( 8 - s ^ { 3 } \right) , & 0 \leq s \leq 2 \\ 0 , & \text { otherwise } \end{cases}$$
4. Find the change in the mean time that competitors stay on the log.

3. In an old computer game a white square representing a ball appears at random at the top of the playing area, which is 24 cm wide, and moves down the screen. The continuous random variable \(X\) represents the distance, in centimetres, of the dot from the left-hand edge of the screen when it appears. The distribution of \(X\) is rectangular over the interval [4,28].

1. Find the mean and variance of \(X\).
2. Find \(\mathrm { P } ( | X - 16 | < 3 )\). During a single game, a player receives 12 "balls".
3. Find the probability that the ball appears within 3 cm of the middle of the top edge of the playing area more than four times in a single game.
   (3 marks)

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

6. \begin{figure}[h] \end{figure} Figure 1 shows a square of side \(t\) and area \(t ^ { 2 }\) which lies in the first quadrant with one vertex at the origin. A point \(P\) with coordinates ( \(X , Y\) ) is selected at random inside the square and the coordinates are used to estimate \(t ^ { 2 }\). It is assumed that \(X\) and \(Y\) are independent random variables each having a continuous uniform distribution over the interval \([ 0 , t ]\).
[0pt] [You may assume that \(\mathrm { E } \left( X ^ { n } Y ^ { n } \right) = \mathrm { E } \left( X ^ { n } \right) \mathrm { E } \left( Y ^ { n } \right)\), where \(n\) is a positive integer.]

1. Use integration to show that \(\mathrm { E } \left( X ^ { n } \right) = \frac { t ^ { n } } { n + 1 }\). The random variable \(S = k X Y\), where \(k\) is a constant, is an unbiased estimator for \(t ^ { 2 }\).
2. Find the value of \(k\).
3. Show that \(\operatorname { Var } S = \frac { 7 t ^ { 4 } } { 9 }\). The random variable \(U = q \left( X ^ { 2 } + Y ^ { 2 } \right)\), where \(q\) is a constant, is also an unbiased estimator for \(t ^ { 2 }\).
4. Show that the value of \(q = \frac { 3 } { 2 }\).
5. Find Var \(U\).
6. State, giving a reason, which of \(S\) and \(U\) is the better estimator of \(t ^ { 2 }\). The point \(( 2,3 )\) is selected from inside the square.
7. Use the estimator chosen in part (f) to find an estimate for the area of the square.

6. Emily is monitoring the level of pollution in a river. Over a period of time she has found that the amount of pollution, \(X\), in a 100 ml sample of river water has a continuous distribution with probability density function \(\mathrm { f } ( x )\) given by $$f ( x ) = \left\{ \begin{array} { c c } \frac { 2 x } { a ^ { 2 } } & 0 \leqslant x \leqslant a \\ 0 & \text { otherwise } \end{array} \right.$$ where \(a\) is a constant. Emily takes a random sample \(X _ { 1 } , X _ { 2 } , X _ { 3 } , \ldots , X _ { n }\) to try to estimate the value of \(a\).

1. Show that \(\mathrm { E } ( \bar { X } ) = \frac { 2 a } { 3 }\) and \(\operatorname { Var } ( \bar { X } ) = \frac { a ^ { 2 } } { 18 n }\) The random variable \(S = p \bar { X }\), where \(p\) is a constant, is an unbiased estimator of \(a\).
2. Write down the value of \(p\) and find \(\operatorname { Var } ( S )\). Felix suggests using the statistic \(M = \max \left\{ X _ { 1 } , X _ { 2 } , X _ { 3 } , \ldots , X _ { n } \right\}\) as an estimator of \(a\).
   He calculates \(\mathrm { E } ( M ) = \frac { 2 n } { 2 n + 1 } a\) and \(\operatorname { Var } ( M ) = \frac { n } { ( n + 1 ) ( 2 n + 1 ) ^ { 2 } } a ^ { 2 }\)
3. State, giving your reasons, whether or not \(M\) is a consistent estimator of \(a\). The random variable \(T = q M\), where \(q\) is a constant, is an unbiased estimator of \(a\).
4. Write down, in terms of \(n\), the value of \(q\) and find \(\operatorname { Var } ( T )\).
5. State, giving your reasons, which of \(S\) or \(T\) you would recommend Emily use as an estimator of \(a\). Emily took a sample of 5 values of \(X\) and obtained the following:
   5.3
   4.3 \(\begin{array} { l l } 5.7 & 7.8 \end{array}\) 6.9
6. Calculate the estimate of \(a\) using your recommended estimator from part (e).
7. Find the standard error of your estimate, giving your answer to 2 decimal places.

6. A random sample \(X _ { 1 } , X _ { 2 } , X _ { 3 } , \ldots , X _ { 2 n }\) is taken from a population with mean \(\frac { \mu } { 3 }\) and variance \(3 \sigma ^ { 2 }\). A second random sample \(Y _ { 1 } , Y _ { 2 } , Y _ { 3 } , \ldots , Y _ { n }\) is taken from a population with mean \(\frac { \mu } { 2 }\) and variance \(\frac { \sigma ^ { 2 } } { 2 }\), where the \(X\) and \(Y\) variables are all independent. \(A\), \(B\) and \(C\) are possible estimators of \(\mu\), where $$\begin{aligned} & A = \frac { X _ { 1 } + X _ { 2 } + X _ { 3 } + Y _ { 1 } + Y _ { 2 } } { 2 } \\ & B = \frac { 3 X _ { 1 } } { 2 } + \frac { 2 Y _ { 1 } } { 3 } \\ & C = \frac { 3 X _ { 1 } + 4 Y _ { 1 } } { 3 } \end{aligned}$$

1. Show that two of \(A , B\) and \(C\) are unbiased estimators of \(\mu\) and find the bias of the third estimator of \(\mu\).
2. Showing your working clearly, find which of \(A\), \(B\) and \(C\) is the best estimator of \(\mu\). The estimator $$D = \frac { 1 } { k } \left( \sum _ { i = 1 } ^ { 2 n } X _ { i } + \sum _ { i = 1 } ^ { n } Y _ { i } \right)$$ is an unbiased estimator of \(\mu\).
3. Find \(k\) in terms of \(n\).
4. Show that \(D\) is also a consistent estimator of \(\mu\).
5. Find the least value of \(n\) for which \(D\) is a better estimator of \(\mu\) than any of \(A\), \(B\) or \(C\).

6. A random sample of size \(n\) is taken from the random variable \(X\), which has a continuous uniform distribution over the interval [ \(0 , a\) ], \(a > 0\) The sample mean is denoted by \(\bar { X }\)

1. Show that \(Y = 2 \bar { X }\) is an unbiased estimator of \(a\) The maximum value, \(M\), in the sample has probability density function $$f ( m ) = \left\{ \begin{array} { c c } \frac { n m ^ { n - 1 } } { a ^ { n } } & 0 \leqslant m \leqslant a \\ 0 & \text { otherwise } \end{array} \right.$$
2. Find E(M)
3. Show that \(\operatorname { Var } ( M ) = \frac { n a ^ { 2 } } { ( n + 2 ) ( n + 1 ) ^ { 2 } }\) The estimator \(S\) is defined by \(S = \frac { n + 1 } { n } M\) Given that \(n > 1\)
4. state which of \(Y\) or \(S\) is the better estimator for \(a\). Give a reason for your answer.

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

$$f ( x ) = \left\{ \begin{array} { c c } \frac { x } { 2 \theta ^ { 2 } } & 0 \leqslant x \leqslant 2 \theta \\ 0 & \text { otherwise } \end{array} \right.$$ where \(\theta\) is a constant.

1. Use integration to show that \(\mathrm { E } \left( X ^ { N } \right) = \frac { 2 ^ { N + 1 } } { N + 2 } \theta ^ { N }\)
2. Hence
   1. write down an expression for \(\mathrm { E } ( X )\) in terms of \(\theta\)
   2. find \(\operatorname { Var } ( X )\) in terms of \(\theta\) A random sample \(X _ { 1 } , X _ { 2 } , \ldots , X _ { n }\) where \(n \geqslant 2\) is taken to estimate the value of \(\theta\) The random variable \(S _ { 1 } = q \bar { X }\) is an unbiased estimator of \(\theta\)
3. Write down the value of \(q\) and show that \(S _ { 1 }\) is a consistent estimator of \(\theta\) The continuous random variable \(Y\) is independent of \(X\) and is uniformly distributed over the interval \(\left[ 0 , \frac { 2 \theta } { 3 } \right]\), where \(\theta\) is the same unknown constant as in \(\mathrm { f } ( x )\). The random variable \(S _ { 2 } = a X + b Y\) is an unbiased estimator of \(\theta\) and is based on one observation of \(X\) and one observation of \(Y\).
4. Find the value of \(a\) and the value of \(b\) for which \(S _ { 2 }\) has minimum variance.
5. Show that the minimum variance of \(S _ { 2 }\) is \(\frac { \theta ^ { 2 } } { 11 }\)
6. Explain which of \(S _ { 1 }\) or \(S _ { 2 }\) is the better estimator for \(\theta\)

3 The probability density function of the continuous random variable \(X\) is given by $$\mathrm { f } ( x ) = \begin{cases} c + x & - c \leqslant x \leqslant 0 \\ c - x & 0 \leqslant x \leqslant c \\ 0 & \text { otherwise } \end{cases}$$ where \(c\) is a positive constant.

1. (A) Sketch the graph of the probability density function.
   (B) Show that \(c = 1\).
2. Find \(\mathrm { P } \left( X < \frac { 1 } { 4 } \right)\).
3. Find

4 The random variable \(X\) has a continuous uniform distribution on [ 0,10 ].

1. Find \(\mathrm { P } ( 3 < X < 6 )\).
2. Find each of the following.
   Marisa is investigating the sample mean, \(Y\), of 8 independent values of \(X\). She designs a simulation shown in the spreadsheet in Fig. 4.1. Each of the 25 rows below the heading row consists of 8 values of \(X\) together with the value of \(Y\). All of the values in the spreadsheet have been rounded to 2 decimal places. \begin{table}[h]

   1	A	B	C	D	E	F	G	H	I	J
   1	\(X _ { 1 }\)	\(X _ { 2 }\)	\(X _ { 3 }\)	\(X _ { 4 }\)	\(X _ { 5 }\)	\(X _ { 6 }\)	\(X _ { 7 }\)	\(X _ { 8 }\)	\(Y\)
   2	6.31	2.45	3.27	3.06	4.16	1.53	0.43	7.99	3.65
   3	1.70	1.52	7.10	8.93	6.44	2.70	9.96	7.83	5.77
   4	9.15	0.52	4.95	6.99	6.52	3.15	0.81	5.35	4.68
   5	0.65	2.71	7.92	9.65	0.50	4.87	6.46	2.67	4.43
   6	3.09	6.11	3.96	0.09	0.18	4.67	0.67	6.20	3.12
   7	7.06	5.84	1.97	3.60	9.36	1.97	4.48	3.47	4.72
   8	1.46	1.57	5.45	0.37	3.76	7.56	8.48	9.12	4.72
   9	9.42	1.85	4.91	1.61	1.94	8.00	1.77	5.34	4.36
   10	2.98	5.32	2.91	4.12	9.16	1.76	9.97	6.88	5.39
   11	2.83	3.44	3.28	7.85	1.00	0.93	8.77	4.03	4.01
   12	4.51	0.59	5.84	9.87	8.65	3.94	7.18	0.23	5.10
   13	4.49	0.69	3.65	8.78	4.96	8.96	3.77	1.43	4.59
   14	6.57	8.08	4.85	6.75	7.92	0.27	9.69	4.04	6.02
   15	8.35	1.09	8.63	8.04	7.23	2.12	2.57	9.59	5.95
   16	5.24	9.53	6.08	8.21	3.61	7.07	6.65	7.63	6.75
   17	7.89	5.50	3.09	0.71	6.47	5.49	6.47	4.95	5.07
   18	8.36	7.27	2.35	9.04	0.58	2.26	3.01	7.90	5.10
   19	3.76	1.01	9.61	9.65	7.89	9.98	6.28	4.34	6.56
   20	9.94	6.84	3.38	5.53	0.26	8.53	5.72	5.12	5.66
   21	7.25	9.10	0.34	2.88	4.66	2.65	6.37	7.63	5.11
   22	7.18	7.14	5.38	0.04	4.09	6.47	4.96	4.23	4.94
   23	8.69	5.04	4.90	2.94	2.00	4.23	4.13	0.97	4.11
   24	3.46	6.33	0.48	9.35	0.23	1.18	7.97	6.37	4.42
   25	2.37	7.26	7.16	1.24	5.26	2.80	3.55	3.84	4.19
   26	2.16	8.30	7.17	3.32	2.96	1.30	9.11	0.31	4.33
   27

   \captionsetup{labelformat=empty} \caption{Fig. 4.1}
   \end{table}
3. Use the spreadsheet to estimate \(\mathrm { P } ( 3 < Y < 6 )\).
4. Explain why it is not surprising that this estimated probability is substantially greater than the value which you calculated in part (i). Marisa wonders whether, even though the sample size is only 8, use of the Central Limit Theorem will provide a good approximation to \(\mathrm { P } ( 3 < Y < 6 )\).
5. Calculate an estimate of \(\mathrm { P } ( 3 < Y < 6 )\) using the Central Limit Theorem. A Normal probability plot of the 25 simulated values of \(Y\) is shown in Fig. 4.2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{0c58d4d7-10e9-473a-888a-b407ec90bf08-5_800_1291_306_386} \captionsetup{labelformat=empty} \caption{Fig. 4.2}
   \end{figure}
6. Explain what the Normal probability plot suggests about the use of the Central Limit Theorem to approximate \(\mathrm { P } ( 3 < Y < 6 )\). Marisa now decides to use a spreadsheet with 1000 rows below the heading row, rather than the 25 which she used in the initial simulation shown in Fig. 4.1. She uses a counter to count the number of values of \(Y\) between 3 and 6. This value is 808.
7. Explain whether the value 808 supports the suggestion that the Central Limit Theorem provides a good approximation to \(\mathrm { P } ( 3 < Y < 6 )\). Marisa decides to repeat each of her two simulations many times in order to investigate how variable the probability estimates are in each case.
8. Explain whether you would expect there to be more, the same or less variability in the probability estimates based on 1000 rows than in the probability estimates based on 25 rows.

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 continuous random variable \(X\) has cumulative distribution function given by \(F ( x ) = \begin{cases} 0 & x < a , \\ \frac { x - a } { b - a } & a \leqslant x \leqslant b , \\ 1 & x > b , \end{cases}\) where \(a\) and \(b\) are constants with \(0 < \mathrm { a } < \mathrm { b }\).

1. Find \(\mathrm { P } \left( \mathrm { X } < \frac { 1 } { 2 } ( \mathrm { a } + \mathrm { b } ) \right)\).
2. Sketch the graph of the probability density function of \(X\).
3. Find the variance of \(X\) when \(a = 2\) and \(b = 8\).

7 Many cars have pollen filters to try to remove as much pollen as possible from the passenger compartment. In a test car, the amount of pollen is regularly monitored. The amount of pollen is measured using a scale from 0 to 1 , and is modelled by the continuous random variable \(X\) with probability density function given by \(f ( x ) = \begin{cases} k \left( 5 x ^ { 4 } - 16 x ^ { 2 } + 11 x \right) & 0 \leqslant x \leqslant 1 , \\ 0 & \text { otherwise } , \end{cases}\) where \(k\) is a positive constant.

1. Show that \(k = \frac { 6 } { 7 }\).
2. Determine \(\mathrm { P } ( X < \mathrm { E } ( X ) )\).
3. Verify that the median amount of pollen according to the model lies between 0.417 and 0.418.

6 The probability density function of the continuous random variable \(X\) is given by \(f ( x ) = \begin{cases} 2 ( 1 + a x ) & 0 \leqslant x \leqslant 1 , \\ 0 & \text { otherwise } , \end{cases}\) where \(a\) is a constant.

1. Show that \(a = - 1\).
2. Find the cumulative distribution function of \(X\).
3. Find \(\mathrm { P } ( X < 0.5 )\).
4. Show that \(\mathrm { E } ( X )\) is greater than the median of \(X\).

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 \(Y\).

3 At a factory, flour is packed into bags. A model for the mass in grams of flour packed into each bag is \(1500 + X\), where \(X\) is a continuous random variable with probability density function $$f ( x ) = \left\{ \begin{array} { c c } k x ( 6 - x ) & 0 \leq x \leq 6 \\ 0 & \text { elsewhere, } \end{array} \right.$$ where \(k\) is a constant.

1. Show that \(k = \frac { 1 } { 36 }\).
2. Find the probability that a randomly selected bag of flour contains 1505 grams of flour or more.
3. Find

6 A lottery has tickets numbered 1 to \(n\) inclusive, where \(n\) is a positive integer. The random variable \(X\) denotes the number on a ticket drawn at random.

1. Determine \(\mathrm { P } \left( \mathrm { X } \leqslant \frac { 1 } { 4 } \mathrm { n } \right)\) in each of the following cases.
   1. \(n\) is a multiple of 4 .
   2. \(n\) is of the form \(4 k + 1\), where \(k\) is a positive integer. Give your answer as a single fraction in terms of \(n\).
2. Given that \(n = 101\), find the probability that \(X\) is within one standard deviation of the mean.