Questions S1 - A-Level Maths

The birth weights, $W$ grams, of a particular breed of kitten are assumed to be normally distributed with mean 99 g and standard deviation 3.6 g
1. Find $\mathrm { P } ( W > 92 )$
2. Find, to one decimal place, the value of $k$ such that $\mathrm { P } ( W < k ) = 3 \mathrm { P } ( W > k )$
3. Write down the name given to the value of $k$.
For a different breed of kitten, the birth weights are assumed to be normally distributed with mean 120 g Given that the 20th percentile for this breed of kitten is 116 g
find the standard deviation of the birth weight of this breed of kitten.

Edexcel S1 2016 January Q1

12 marks Moderate -0.8

The discrete random variable $X$ has the probability distribution given in the table below.

$x$	- 2	1	3	4	6
$\mathrm { P } ( X = x )$	$\frac { 1 } { 4 }$	$\frac { 1 } { 6 }$	$\frac { 1 } { 3 }$	$\frac { 1 } { 12 }$	$\frac { 1 } { 6 }$

Write down the value of $\mathrm { F } ( 5 )$
Find $\mathrm { E } ( X )$
Find $\operatorname { Var } ( X )$ The random variable $Y = 7 - 2 X$
Find
1. $\mathrm { E } ( Y )$
2. $\operatorname { Var } ( Y )$
3. $\mathrm { P } ( Y > X )$ \includegraphics[max width=\textwidth, alt={}, center]{70137e9a-0a6b-48b5-8dd4-c436cb063351-03_2261_47_313_37}

Edexcel S1 2016 January Q2

12 marks Easy -1.2

2. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{70137e9a-0a6b-48b5-8dd4-c436cb063351-04_284_1244_260_388} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows part of a box and whisker plot for the marks in an examination with a large number of candidates. Part of the lower whisker has been torn off.

Given that $75 \%$ of the candidates passed the examination, state the lowest mark for the award of a pass.
Given that the top $25 \%$ of the candidates achieved a merit grade, state the lowest mark for the award of a merit grade. An outlier is defined as any value greater than $c$ or any value less than $d$ where $$\begin{aligned} & c = Q _ { 3 } + 1.5 \left( Q _ { 3 } - Q _ { 1 } \right) \\ & d = Q _ { 1 } - 1.5 \left( Q _ { 3 } - Q _ { 1 } \right) \end{aligned}$$
Find the value of $c$ and the value of $d$.
Write down the 3 highest marks scored in the examination. The 3 lowest marks in the examination were 5, 10 and 15
On the diagram on page 7, complete the box and whisker plot. Three candidates are selected at random from those who took this examination.
Find the probability that all 3 of these candidates passed the examination but only 2 achieved a merit grade.
\includegraphics[max width=\textwidth, alt={}, center]{70137e9a-0a6b-48b5-8dd4-c436cb063351-05_285_1628_2343_166} Turn over for a spare diagram if you need to redraw your plot.

Edexcel S1 2016 January Q3

15 marks Moderate -0.3

3. A publisher collects information about the amount spent on advertising, $\pounds x$, and the sales, $y$ books, for some of her publications. She collects information for a random sample of 8 textbooks and codes the data using $v = \frac { x + 50 } { 200 }$ and $s = \frac { y } { 1000 }$ to give

$v$	0.60	8.10	4.30	0.40	1.60	6.40	2.50	5.10
$s$	1.84	6.73	5.95	1.30	2.45	7.46	4.82	6.25

[You may use: $\sum v = 29 \sum s = 36.8 \sum s ^ { 2 } = 209.72 \sum v s = 177.311 \quad \mathrm {~S} _ { v v } = 55.275$ ]

Find $\mathrm { S } _ { v s }$ and $\mathrm { S } _ { s s }$
Calculate the product moment correlation coefficient for these data. The publisher believes that a linear regression model may be appropriate to describe these data.
State, giving a reason, whether or not your answer to part (b) supports the publisher's belief.
Find the equation of the regression line of $s$ on $v$, giving your answer in the form $s = a + b v$
Hence find the equation of the regression line of $y$ on $x$ for the sample of textbooks, giving your answer in the form $y = c + d x$ The publisher calculated the regression line for a sample of novels and obtained the equation $$y = 3100 + 1.2 x$$ She wants to increase the sales of books by spending more money on advertising.
State, giving your reasons, whether the publisher should spend more money on advertising textbooks or novels.

Edexcel S1 2016 January Q4

13 marks Moderate -0.3

4. A training agency awards a certificate to each student who passes a test while completing a course.
Students failing the test will attempt the test again up to 3 more times, and, if they pass the test, will be awarded a certificate.
The probability of passing the test at the first attempt is 0.7 , but the probability of passing reduces by 0.2 at each attempt.

Complete the tree diagram below to show this information.
\includegraphics[max width=\textwidth, alt={}, center]{70137e9a-0a6b-48b5-8dd4-c436cb063351-08_545_1244_639_340} A student who completed the course is selected at random.
Find the probability that the student was awarded a certificate.
Given that the student was awarded a certificate, find the probability that the student passed on the first or second attempt. The training agency decides to alter the test taken by the students while completing the course, but will not allow more than 2 attempts. The agency requires the probability of passing the test at the first attempt to be $p$, and the probability of passing the test at the second attempt to be ( $p - 0.2$ ). The percentage of students who complete the course and are awarded a certificate is to be $95 \%$
Show that $p$ satisfies the equation $$p ^ { 2 } - 2.2 p + 1.15 = 0$$
Hence find the value of $p$, giving your answer to 3 decimal places.
\includegraphics[max width=\textwidth, alt={}, center]{70137e9a-0a6b-48b5-8dd4-c436cb063351-09_2261_47_313_37}

Edexcel S1 2016 January Q5

14 marks Standard +0.3

5. Rosie keeps bees. The amount of honey, in kg, produced by a hive of Rosie's bees in a season, is modelled by a normal distribution with a mean of 22 kg and a standard deviation of 10 kg .

Find the probability that a hive of Rosie's bees produces less than 18 kg of honey in a season. The local bee keepers’ club awards a certificate to every hive that produces more than 39 kg of honey in a season, and a medal to every hive that produces more than 50 kg in a season. Given that one of Rosie's bee hives is awarded a certificate
find the probability that this hive is also awarded a medal.
(5) Sam also keeps bees. The amount of honey, in kg, produced by a hive of Sam's bees in a season, is modelled by a normal distribution with mean $\mu \mathrm { kg }$ and standard deviation $\sigma \mathrm { kg }$. The probability that a hive of Sam’s bees produces less than 28 kg of honey in a season is 0.8413 Only 20\% of Sam's bee hives produce less than 18 kg of honey in a season.
Find the value of $\mu$ and the value of $\sigma$. Give your answers to 2 decimal places.
(6)

\includegraphics[max width=\textwidth, alt={}, center]{70137e9a-0a6b-48b5-8dd4-c436cb063351-11_2261_47_313_37}

Edexcel S1 2016 January Q6

9 marks Moderate -0.8

6. Yujie is investigating the weights of 10 young rabbits. She records the weight, $x$ grams, of each rabbit and the results are summarised below. $$\sum x = 8360 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 63840$$

Calculate the mean and the standard deviation of the weights of these rabbits. Given that the median weight of these rabbits is 815 grams,
describe, giving a reason, the skewness of these data. Two more rabbits weighing 776 grams and 896 grams are added to make a group of 12 rabbits.
State, giving a reason, how the inclusion of these two rabbits would affect the mean.
By considering the change in $\sum ( x - \bar { x } ) ^ { 2 }$, state what effect the inclusion of these two rabbits would have on the standard deviation.
END

Edexcel S1 2017 January Q1

12 marks Easy -1.3

Ralph records the weights, in grams, of 100 tomatoes. This information is displayed in the histogram below.
\includegraphics[max width=\textwidth, alt={}, center]{1130517e-33d0-41b1-9303-2d981379954d-02_981_1268_338_274}

Given that 5 of the tomatoes have a weight between 2 and 3 grams,

find the number of tomatoes with a weight between 0 and 2 grams. One of the tomatoes is selected at random.
Find the probability that it weighs more than 3 grams.
Estimate the proportion of the tomatoes with a weight greater than 6.25 grams.
Using your answer to part (c), explain whether or not the median is greater than 6.25 grams. Given that the mean weight of these tomatoes is 6.25 grams and using your answer to part (d),
describe the skewness of the distribution of the weights of these tomatoes. Give a reason for your answer. Two of these 100 tomatoes are selected at random.
Estimate the probability that both tomatoes weigh within 0.75 grams of the mean.

Edexcel S1 2017 January Q2

9 marks Easy -1.2

An integer is selected at random from the integers 1 to 50 inclusive.
$A$ is the event that the integer selected is prime.
$B$ is the event that the integer selected ends in a 3
$C$ is the event that the integer selected is greater than 20
The Venn diagram shows the number of integers in each region for the events $A , B$ and $C$
\includegraphics[max width=\textwidth, alt={}, center]{1130517e-33d0-41b1-9303-2d981379954d-04_607_1125_593_413}
1. Describe in words the event $( A \cap B )$
2. Write down the probability that the integer selected is prime.
3. Find $\mathrm { P } \left( [ A \cup B \cup C ] ^ { \prime } \right)$
Given that the integer selected is greater than 20
find the probability that it is prime. Using your answers to (b) and (d),
state, with a reason, whether or not the events $A$ and $C$ are statistically independent. Given that the integer selected is greater than 20 and prime,
find the probability that it ends in a 3

Edexcel S1 2017 January Q3

17 marks Moderate -0.3

A scientist measured the salinity of water, $x \mathrm {~g} / \mathrm { kg }$, and recorded the temperature at which the water froze, $y ^ { \circ } \mathrm { C }$, for 12 different water samples. The summary statistics are listed below.

$$\begin{gathered} \sum x = 504 \quad \sum y = - 27 \quad \sum x ^ { 2 } = 22842 \quad \sum y ^ { 2 } = 62.98 \\ \sum x y = - 1190.7 \quad \mathrm {~S} _ { x x } = 1674 \quad \mathrm {~S} _ { y y } = 2.23 \end{gathered}$$

Find the mean and variance of the recorded temperatures.
(3) Priya believes that the higher the salinity of water, the higher the temperature at which the water freezes.
1. Calculate the product moment correlation coefficient between $x$ and $y$
2. State, with a reason, whether or not this value supports Priya's belief.
Find the least squares regression line of $y$ on $x$ in the form $y = a + b x$ Give the value of $a$ and the value of $b$ to 3 significant figures.
Estimate the temperature at which water freezes when the salinity is $32 \mathrm {~g} / \mathrm { kg }$ The coding $w = 1.8 y + 32$ is used to convert the recorded temperatures from ${ } ^ { \circ } \mathrm { C }$ to ${ } ^ { \circ } \mathrm { F }$
Find an equation of the least squares regression line of $w$ on $x$ in the form $w = c + d x$
Find
1. the variance of the recorded temperatures when converted to ${ } ^ { \circ } \mathrm { F }$
2. the product moment correlation coefficient between $w$ and $x$
  \href{http://PhysicsAndMathsTutor.com}{PhysicsAndMathsTutor.com}

Edexcel S1 2017 January Q4

13 marks Moderate -0.3

In a game, the number of points scored by a player in the first round is given by the random variable $X$ with probability distribution

$x$	5	6	7	8
$\mathrm { P } ( X = x )$	0.13	0.21	0.29	0.37

Find

$\mathrm { E } ( X )$
$\operatorname { Var } ( X )$
$\operatorname { Var } ( 3 - 2 X )$ The number of points scored by a player in the second round is given by the random variable $Y$ and is independent of the number of points scored in the first round. The random variable $Y$ has probability function $$\mathrm { P } ( Y = y ) = \frac { 1 } { 4 } \quad \text { for } y = 5,6,7,8$$
Write down the value of $\mathrm { E } ( Y )$
Find $\mathrm { P } ( X = Y )$
Find the probability that the number of points scored by a player in the first round is greater than the number of points scored by the player in the second round.

Edexcel S1 2017 January Q5

6 marks Moderate -0.3

In a survey, people were asked if they use a computer every day.

Of those people under 50 years old, $80 \%$ said they use computer every day. Of those people aged 50 or more, $55 \%$ said they use computer every day. The proportion of people in the survey under 50 years old is $p$

Draw a tree diagram to represent this information. In the survey, 70\% of all people said they use computer every day.
Find the value of $p$ One person is selected at random. Given that this person uses a computer every day,
find the probability that this person is under 50 years old.
\href{http://PhysicsAndMathsTutor.com}{PhysicsAndMathsTutor.com}

Edexcel S1 2017 January Q6

8 marks Moderate -0.3

In a factory, a machine is used to fill bags of rice. The weights of bags of rice are modelled using a normal distribution with mean 250 g .

Only $1 \%$ of the bags of rice weigh more than 256 g .

Write down the percentage of bags of rice with weights between 244 g and 256 g .
Find the standard deviation of the weights of the bags of rice. An inspection consists of selecting a bag of rice at random and checking if its weight is within 4 g of the mean. If the weight is more than 4 g away from the mean, then a second bag of rice is selected at random and checked. If the weight of each of the 2 bags of rice is more than 4 g away from the mean, then the machine is shut down.
Find the probability that the machine is shut down following an inspection.

Edexcel S1 2017 January Q7

10 marks Standard +0.3

The discrete random variable $X$ can take only the values $1,2,3$ and 4 . For these values, the probability function is given by

$$\mathrm { P } ( X = x ) = \frac { a x + b } { 60 } \quad \text { for } x = 1,2,3,4$$ where $a$ and $b$ are constants.

Show that $5 a + 2 b = 30$ Given that $\mathrm { F } ( 3 ) = \frac { 13 } { 20 }$
find the value of $a$ and the value of $b$ Given also that $Y = X ^ { 2 }$
find the cumulative distribution function of $Y$

Edexcel S1 2018 January Q1

12 marks Moderate -0.8

Two classes of students, class $A$ and class $B$, sat a test.

Class $A$ has 10 students. Class $B$ has 15 students. Each student achieved a score, $x$, on the test and their scores are summarised in the table below.

\cline { 2 - 4 } \multicolumn{1}{c\|}{}	$n$	$\sum x$	$\sum x ^ { 2 }$
Class $A$	10	770	59610
Class $B$	15	$t$	58035

The mean score for Class $A$ is 77 and the mean score for Class $B$ is 61

Find the value of $t$
Calculate the variance of the test scores for each class. The highest score on the test was 95 and the lowest score was 45 These were each scored by students from the same class.
State, with a reason, which class you believe they were from. The two classes are combined into one group of 25 students.
1. Find the mean test score for all 25 students.
2. Find the variance of the test scores for all 25 students. The teacher of class $A$ later realises that he added up the test scores for his class incorrectly. Each student's test score in class $A$ should be increased by 3
Without further calculations, state, with a reason, the effect this will have on
1. the variance of the test scores for class $A$
2. the mean test score for all 25 students
3. the variance of the test scores for all 25 students.

Edexcel S1 2018 January Q2

8 marks Moderate -0.8

2. (a) Shade the region representing the event $A \cup B ^ { \prime }$ on the Venn diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{01259350-0119-4500-a81b-bfa1b4234559-06_355_563_306_694} The two events $C$ and $D$ are mutually exclusive.
Given that $\mathrm { P } ( C ) = \frac { 1 } { 5 }$ and $\mathrm { P } ( D ) = \frac { 3 } { 10 }$ find
(b) (i) $\quad \mathrm { P } ( C \cup D )$
(ii) $\mathrm { P } ( C \mid D )$ The two events $F$ and $G$ are independent.
Given that $\mathrm { P } ( F ) = \frac { 1 } { 6 }$ and $\mathrm { P } ( F \cup G ) = \frac { 3 } { 8 }$ find
(c) (i) $\mathrm { P } ( G )$
(ii) $\mathrm { P } \left( F \mid G ^ { \prime } \right)$

Edexcel S1 2018 January Q3

8 marks Moderate -0.8

3. Martin is investigating the relationship between a person's daily caffeine consumption, $c$ milligrams, and the amount of sleep they get, $h$ hours, per night. He collected this information from 20 people and the results are summarised below. $$\begin{array} { c c } \sum c = 3660 \quad \sum h = 126 \quad \sum c ^ { 2 } = 973228 \\ \sum c h = 20023.4 \quad S _ { c c } = 303448 \quad S _ { c h } = - 3034.6 \end{array}$$ Martin calculates the product moment correlation coefficient for these data and obtains - 0.833

Give a reason why this value supports a linear relationship between $c$ and $h$ The amount of sleep per night is the response variable.
Explain what you understand by the term 'response variable'. Martin says that for each additional 100 mg of caffeine consumed, the expected number of hours of sleep decreases by 1
Determine, by calculation, whether or not the data support this statement.
Use the data to calculate an estimate for the expected number of hours of sleep per night when no caffeine is consumed.

Edexcel S1 2018 January Q4

13 marks

4. The discrete random variable $X$ has probability distribution

$x$	- 4	- 3	1	2	5
$\mathrm { P } ( X = x )$	$a$	$b$	$a$	$b$	0.2

Find $\mathrm { E } ( X )$ in terms of $a$ and $b$ For this probability distribution, $\operatorname { Var } ( X ) = \mathrm { E } \left( X ^ { 2 } \right)$
1. Write down the value of $\mathrm { E } ( X )$
2. Find the value of $a$ and the value of $b$
Find $\operatorname { Var } ( 1 - 3 X )$ Given that $Y = 1 - X$, find
1. $\mathrm { P } ( Y < 0 )$
2. the largest possible value of $k$ such that $\mathrm { P } ( Y < k ) = 0.2$

Edexcel S1 2018 January Q5

12 marks Moderate -0.3

5. Franca is the manager of an accountancy firm. She is investigating the relationship between the salary, $\pounds x$, and the length of commute, $y$ minutes, for employees at the firm. She collected this information from 9 randomly selected employees. The salary of each employee was then coded using $w = \frac { x - 20000 } { 1000 }$ The table shows the values of $w$ and $y$ for the 9 employees.

$w$	6	8	8	- 1	25	15	3	- 2	19
$y$	45	50	35	65	25	40	50	75	20

(You may use $\sum w = 81 \quad \sum y = 405 \quad \sum w y = 2490 \quad S _ { w w } = 660 \quad S _ { y y } = 2500$ )

Calculate the salary of the employee with $w = - 2$
Show that, to 3 significant figures, the value of the product moment correlation coefficient between $w$ and $y$ is - 0.899
State, giving a reason, the value of the product moment correlation coefficient between $x$ and $y$ The least squares regression line of $y$ on $w$ is $y = 60.75 - 1.75 w$
Find the equation of the least squares regression line of $y$ on $x$ giving your answer in the form $y = a + b x$
Estimate the length of commute for an employee with a salary of $\pounds 21000$ Franca uses the regression line to estimate the length of commute for employees with salaries between $\pounds 25000$ and $\pounds 40000$
State, giving a reason, whether or not these estimates are reliable.

Edexcel S1 2018 January Q6

11 marks Standard +0.3

Anju has a bag that contains 5 socks of which 2 are blue.

Anju randomly selects socks from the bag, one sock at a time. She does not replace any socks but continues to select socks at random until she has both blue socks. The discrete random variable $S$ represents the total number of socks that Anju has selected.

Write down the value of $\mathrm { P } ( S = 1 )$
Find $\mathrm { P } ( S > 2 )$
Find $\mathrm { P } ( S = 3 )$
Given that the second sock selected is blue, find the probability that Anju selects exactly 3 socks.
Find $\mathrm { P } ( S = 5 )$

Edexcel S1 2018 January Q7

11 marks Standard +0.3

7. The weights, $G$, of a particular breed of gorilla are normally distributed with mean 180 kg and standard deviation 15 kg .

Find the proportion of these gorillas whose weights exceed 174 kg .
Find, to 1 decimal place, the value of $k$ such that $\mathrm { P } ( k < G < 174 ) = 0.3196$ The weights, $B$, of a particular breed of buffalo are normally distributed with mean 216 kg and standard deviation 30 kg . Given that $\mathrm { P } ( G > w ) = \mathrm { P } ( B < w ) = p$
1. find the value of $w$
2. find the value of $p$ and standard deviation 15 kg .
Find the proportion of these gorillas whose weights exceed 174 kg .
Find, to 1 decimal place, the value of $k$ such that $\mathrm { P } ( k < G < 174 ) = 0.3196$
Leave blank
Q7

\hline &
\hline \end{tabular}

Edexcel S1 2019 January Q1

9 marks Standard +0.3

The Venn diagram shows the probability of a randomly selected student from a school being in the sets $L , B$ and $C$, where
$L$ represents the event that the student has instrumental music lessons
$B$ represents the event that the student plays in the school band
$C$ represents the event that the student sings in the school choir
$p , q , r$ and $s$ are probabilities.
\includegraphics[max width=\textwidth, alt={}, center]{d3f4450d-60eb-49b6-be1b-d2fcfad0451f-02_504_750_735_598}
1. Select a pair of mutually exclusive events from $L , B$ and $C$.
Given that $\mathrm { P } ( L ) = 0.4 , \mathrm { P } ( B ) = 0.13 , \mathrm { P } ( C ) = 0.3$ and the events $L$ and $C$ are independent,
find the value of $p$,
find the value of $q$, the value of $r$ and the value of $s$. A student is selected at random from those who play in the school band or sing in the school choir.
Find the exact probability that this student has instrumental music lessons.

Edexcel S1 2019 January Q2

10 marks Standard +0.3

2. The discrete random variable $X$ has the following probability distribution.

$x$	- 2	- 1	0	1	3
$\mathrm { P } ( X = x )$	0.15	$a$	$b$	$a$	0.4

Find $\mathrm { E } ( X )$. Given that $\mathrm { E } \left( X ^ { 2 } \right) = 4.54$
find the value of $a$ and the value of $b$. The random variable $Y = 3 - 2 X$
Find $\operatorname { Var } ( Y )$.

Edexcel S1 2019 January Q3

10 marks Standard +0.3

The weights of women boxers in a tournament are normally distributed with mean 64 kg and standard deviation 8 kg .
1. Find the probability that a randomly chosen woman boxer in the tournament weighs less than 51 kg .
In the tournament, women boxers who weigh less than 51 kg are classified as lightweight. Ren weighs 49 kg and she has a match against another randomly selected, lightweight woman boxer.
Find the probability that Ren weighs less than the other boxer. In the tournament, women boxers who weigh more than $H \mathrm {~kg}$ are classified as heavyweight. Given that $10 \%$ of the women boxers in the tournament are classified as heavyweight,
find the value of $H$.

Edexcel S1 2019 January Q4

13 marks

4. A group of 100 adults recorded the amount of time, $t$ minutes, they spent exercising each day. Their results are summarised in the table below.

Time (t minutes)	Frequency (f)	Time midpoint (x)
$0 \leqslant t < 15$	25	7.5
$15 \leqslant t < 30$	17	22.5
$30 \leqslant t < 60$	28	45
$60 \leqslant t < 120$	24	90
$120 \leqslant t \leqslant 240$	6	180

[You may use $\sum \mathrm { f } x ^ { 2 } = 455$ 512.5]
A histogram is drawn to represent these data.
The bar representing the time $0 \leqslant t < 15$ has width 0.5 cm and height 6 cm .

Calculate the width and height of the bar representing a time of $60 \leqslant t < 120$
Use linear interpolation to estimate the median time spent exercising by these adults each day.
Find an estimate of the mean time spent exercising by these adults each day.
Calculate an estimate for the standard deviation of these times.
Describe, giving a reason, the skewness of these data. Further analysis of the above data revealed that 18 of the 25 adults in the $0 \leqslant t < 15$ group took no exercise each day.
State, giving a reason, what effect, if any, this new information would have on your answers to
1. the estimate of the median in part (b),
2. the estimate of the mean in part (c),
3. the estimate of the standard deviation in part (d).

\(x\)	- 2	1	3	4	6
\(\mathrm { P } ( X = x )\)	\(\frac { 1 } { 4 }\)	\(\frac { 1 } { 6 }\)	\(\frac { 1 } { 3 }\)	\(\frac { 1 } { 12 }\)	\(\frac { 1 } { 6 }\)

\cline { 2 - 4 } \multicolumn{1}{c\|}{}	\(n\)	\(\sum x\)	\(\sum x ^ { 2 }\)
Class \(A\)	10	770	59610
Class \(B\)	15	\(t\)	58035

\(x\)	- 4	- 3	1	2	5
\(\mathrm { P } ( X = x )\)	\(a\)	\(b\)	\(a\)	\(b\)	0.2

\(x\)	- 2	- 1	0	1	3
\(\mathrm { P } ( X = x )\)	0.15	\(a\)	\(b\)	\(a\)	0.4

\(v\)	0.60	8.10	4.30	0.40	1.60	6.40	2.50	5.10
\(s\)	1.84	6.73	5.95	1.30	2.45	7.46	4.82	6.25


END

\(w\)	6	8	8	- 1	25	15	3	- 2	19
\(y\)	45	50	35	65	25	40	50	75	20

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