Exercises: Marginal distributions

Given the following two-way table for variables $X$ and $Y$:

$X \setminus Y$	3	4
1	2	5
2	4	1

Write down the table of the marginal distribution of the variable $X$.

The relationship between two statistical variables, $X$ and $Y$, has been studied, obtaining the following joint frequency table:

$X \setminus Y$	10	20
0	3	2
1	1	6
2	4	0

Determine the marginal distribution of the variable $Y$ and calculate the total population size $N$.

The marginal distribution of a two-dimensional variable for its component $X$ is given by the following table:

$x_i$	2	4	6
$n_i$	3	5	2

Calculate the marginal mean $\bar{x}$.

The following pairs of values $(x_i, y_i)$ have been collected from observing 5 individuals:

$(1, 5), (2, 5), (1, 6), (3, 5), (2, 6)$

From these data, construct the table of the marginal distribution of the variable $Y$.

You can do it on paper and send a photo with your answer.

In the following two-way table, the marginal totals (the row and column sums) are missing.

$X \setminus Y$	1	2	3	Total $X$
5	2	0	4	?
6	1	3	1	?
Total $Y$	?	?	?	$N = ?$

Copy the table and fill in the missing marginal values.

The marginal distribution of the variable $Y$ in a statistical study is as follows:

$y_j$	0	1	2
$n_j$	4	4	2

Given that its marginal mean is $\bar{y} = 0,8$, calculate the marginal variance $\sigma_Y^2$.

Look at the following joint frequency table:

$X \setminus Y$	100	200	300
1	5	8	2
2	3	1	7

Identify the value of the marginal frequency $n_{2 \bullet}$ (the total number of individuals with $X=2$) and the value of the marginal frequency $n_{\bullet 3}$ (the total number of individuals with $Y=300$).

In a group of 10 friends, they were asked about the number of siblings ($X$) and the number of pets ($Y$). The results $(X, Y)$ are:

$(1, 1), (2, 0), (1, 1), (0, 2), (2, 1), (1, 0), (3, 0), (1, 1), (0, 1), (2, 0)$

Write down the table of the marginal distribution of the number of pets ($Y$).

In an urban mobility survey conducted by the Transport Consortium, 50 bus users are asked about the number of transfers they make on their regular journey ($X$) and their average waiting time in minutes ($Y$). The results are shown in the following two-way table:

$X \setminus Y$	5	10	15
0	14	6	0
1	5	15	4
2	1	2	3

1. a) Construct the marginal distribution table for the variable $X$ (number of transfers), indicating the values $x_i$ and their respective marginal frequencies $f_{i\bullet}$.
2. b) Calculate the marginal mean $\bar{x}$.
3. c) Interpret what the value of $\bar{x}$ describes in the context of these citizens' regular journeys.

A group of 4th-year ESO students collects data on the number of siblings ($X$) and the weekly hours spent playing video games ($Y$) from a sample of 30 schoolmates:

$X \setminus Y$	2	5	8
0	5	4	3
1	4	6	2
2	2	3	1

Marcos wants to calculate the marginal mean for the number of siblings ($\bar{x}$) and writes in his notebook:

"The possible values of variable $X$ are 0, 1, and 2 siblings. Therefore, the mean is $\bar{x} = \frac{0 + 1 + 2}{3} = \frac{3}{3} = 1$ sibling".

1. a) Clearly explain the conceptual error Marcos made and why his method would always yield the same result regardless of the actual number of students who have 0, 1, or 2 siblings.
2. b) Correct the procedure: obtain the marginal distribution of $X$ with its corresponding frequencies and calculate the true value of the marginal mean $\bar{x}$.

A biologist studies a species of amphibian, recording its mass $X$ (in kg) and its length $Y$ (in cm). The results are grouped in the following two-way table:

$X \setminus Y$	15	20	25
0.5	4	2	0
1.0	1	5	3
1.5	0	2	3

Calculate the marginal means $\bar{x}$ and $\bar{y}$ of both variables.

In a market study on the price of a product $X$ (grouped into intervals of euros) and the number of daily sales $Y$, the following is obtained:

$X \setminus Y$	10	20	30
$[10, 20)$	5	12	3
$[20, 30)$	8	15	7
$[30, 40)$	2	6	12

Obtain the marginal distribution of $X$ using the class midpoints and calculate its marginal mean.

In a basketball team, the number of fouls committed ($X$) and points scored ($Y$) per game are analyzed. The joint frequency table has an unknown frequency $f$:

$X \setminus Y$	5	10	15
1	2	$f$	1
2	3	4	2

Knowing that the marginal mean of points scored is $\bar{y} = 9{,}5$, find the value of the unknown frequency $f$.

The battery life $X$ (in hours) and screen size $Y$ (in inches) of several tablet models are analysed:

$X \setminus Y$	7	8	10
6	2	5	1
8	4	8	3
10	1	4	6
12	0	2	4

Construct the marginal distribution of battery life ($X$) and calculate its marginal standard deviation.

The number of weekly study hours ($X$) and the mark obtained in an exam ($Y$) have been recorded for 10 students:

$(5, 6), (7, 8), (4, 5), (8, 9), (5, 5), (6, 7), (7, 7), (9, 10), (4, 6), (6, 8)$

Construct the two-way table of absolute frequencies. From it, obtain the marginal distribution of the marks ($Y$) and determine what percentage of students obtained a marginal mark higher than 6.

You can do it on paper and send a photo with your answer.

In a clinical trial, the dosage of a medication $X$ (in mg) is related to recovery days $Y$:

$X \setminus Y$	3	5	7
100	6	4	0
200	2	8	2
300	0	3	5

Calculate the marginal coefficient of variation of the variable $Y$ to analyse its relative dispersion.

The mean temperature $X$ (in °C) and monthly rainy days $Y$ are recorded in several cities:

$X \setminus Y$	2	4	6
15	3	5	2
20	4	7	3
25	1	4	6

Extract the marginal distributions of $X$ and $Y$. Verify numerically that the sum of the marginal frequencies of $X$ matches the sum of the marginal frequencies of $Y$, and that both equal the total sample size $N$.

The amount of fertiliser $X$ (in kg/ha) and crop yield $Y$ (in tonnes/ha) are studied in different plots:

$X \setminus Y$	2	3	4
50	5	2	1
100	3	6	2
150	0	4	7

Calculate the marginal variance of $X$ and the marginal variance of $Y$. Which of the two variables shows greater absolute dispersion?

An AEMET weather station in a mountainous area records the minimum temperature in ºC ($X$) and the accumulated snow in centimetres ($Y$) over 40 winter days:

$X \setminus Y$	0	5	10
-2	1	6	8
0	5	12	3
2	4	1	0

1. a) Estimate: Without calculating the marginal mean of the temperature ($\bar{x}$) yet, observe the row marginal frequencies ($f_{i\bullet}$) and reason whether you expect $\bar{x}$ to be a positive number, a negative number, or exactly zero. Justify your prediction by analysing which value of $X$ holds the most weight in the sample.
2. b) Check: Construct the table of the marginal distribution of $X$ and calculate the mean $\bar{x}$ and the marginal variance $\sigma_x^2$ accurately.
3. c) Assess: Do your exact calculations confirm the estimate made in section a)? Explain whether the numerical result obtained makes climatic sense for a winter month in high mountains.

An express parcel delivery company analyses a fleet of $N = 20$ vans according to two variables: the number of assigned districts ($X$, which only takes the values 10 or 20 districts) and the delivery shift ($Y$, where 1 represents the morning shift and 2 represents the afternoon shift).

The individual data from the joint table have been lost, but two marginal conditions are known:

1) The marginal mean of the number of assigned districts is $\bar{x} = 14$.

2) The total number of vans assigned to the morning shift ($Y = 1$) is exactly three times the number of vans assigned to the afternoon shift ($Y = 2$).

1. a) Set up the necessary equations to determine the marginal frequencies of $X$ ($f_{1\bullet}$ and $f_{2\bullet}$) and the marginal frequencies of $Y$ ($f_{\bullet 1}$ and $f_{\bullet 2}$).
2. b) Design a $2 \times 2$ two-way table with non-zero interior joint frequencies that exactly satisfies all the marginal frequencies you calculated.

In a factory, the number of defects $X$ and the inspection time $Y$ (in minutes) of a batch of parts are analysed. The joint frequency table contains two parameters, $a$ and $b$:

$X \setminus Y$	2	4	6
0	10	5	2
1	4	$a$	3
2	1	2	$b$

It is known that the total sample size is $N = 40$ and that the marginal mean of the number of defects is $\bar{x} = 0,8$. Determine the values of the frequencies $a$ and $b$.

In a thermodynamic experiment, pressure $X$ (in atm) and volume $Y$ (in litres) are measured, grouping the data into intervals:

$X \setminus Y$	$[1, 3)$	$[3, 5)$	$[5, 7)$
$[10, 20)$	12	8	0
$[20, 30)$	5	15	6
$[30, 40)$	0	4	10

Extract the marginal distributions using the class midpoints. Calculate their marginal standard deviations and determine which variable has greater relative dispersion by calculating the marginal coefficient of variation.

A study relates income level $X$ (in thousands of euros) and satisfaction index $Y$ (from 1 to 3). Instead of absolute frequencies, the joint relative frequencies $h_{ij}$ are provided:

$X \setminus Y$	1	2	3
15	0.10	0.15	0.05
25	0.05	0.25	0.10
35	0.00	0.10	0.20

Calculate the relative marginal distributions and, from them, obtain the marginal means $\bar{x}$ and $\bar{y}$.

The tensile strength $X$ (in MPa) and the carbon percentage $Y$ are evaluated in steel alloys. The table is incomplete, but the marginal totals are known:

$X \setminus Y$	0,2	0,4	0,6	Total $X$
400	8	?	2	15
500	?	13	?	25
600	1	?	5	10
Total $Y$	12	22	16	$N=50$

Reconstruct the complete two-way table. Then, calculate the marginal variance of $Y$.

The following bubble chart represents the two-dimensional distribution of the Risk Level ($X$) and Expected Return ($Y$) of a set of investment portfolios. The number inside each bubble indicates the joint frequency.

From the chart, extract the marginal distributions of both variables and calculate the marginal mean of the Return ($\bar{y}$).

The age $X$ (in intervals) and the number of children $Y$ of a group of families are analysed:

$X \setminus Y$	0	1	2	3
$[20, 30)$	15	10	2	0
$[30, 40)$	8	18	12	4
$[40, 50)$	3	7	15	6

Calculate the marginal mean and variance of Age ($X$) using the class midpoints. If due to a recording error a correction factor must be applied by adding 2 years to each class midpoint, what will be the new marginal mean and variance of $X$ without having to recalculate the whole table? Give reasons for your answer based on the properties of the mean and variance.

A shop records the base price $X$ (in euros) and the number of units sold $Y$ of different items. The marginal distribution of base prices $X$ obtained from the two-dimensional table is:

$x_i$	10	20	30	40
$n_i$	25	40	20	15

Given that the final retail price ($X'$) is calculated by applying a fixed tax of 5 euros plus a 20% increase on the base price (that is, $X' = 1,20X + 5$), calculate the marginal mean and standard deviation of the final prices $X'$ without having to transform each value individually.

An ecologist measures the amount of sunlight $X$ and the growth of a plant $Y$. The data are summarised in the following table:

$X \setminus Y$	10	20
5	2	3
8	4	1

Prove numerically that calculating the marginal mean of $X$ using the full two-dimensional formula $\bar{x} = \frac{1}{N} \sum_{i} \sum_{j} x_i \cdot n_{ij}$ gives exactly the same result as first extracting the marginal distribution of $X$ and applying the one-dimensional formula $\bar{x} = \frac{1}{N} \sum_{i} x_i \cdot n_{i \bullet}$. Show the step-by-step working for both methods.

In a mechanical components factory, the length in centimetres ($X$) of a part and its wear resistance ($Y$) are monitored. From the marginal distribution of $X$, the quality department has already determined that the mean is $\bar{x} = 12\text{ cm}$ and its standard deviation is $\sigma_x = 0,4\text{ cm}$ (so its variance is $\sigma_x^2 = 0,16\text{ cm}^2$).

To adapt the blueprints to international standards, the length must be expressed in millimetres and a fixed assembly margin of 5 mm must be added, applying the linear transformation $X' = 10 \cdot X + 5$.

To find the new marginal mean $\bar{x}'$ and the new marginal variance $\sigma_{x'}^2$, two methods are proposed:

• Method 1: Transform each original individual value $x_i$ by calculating $x'_i = 10 \cdot x_i + 5$, rebuild the entire marginal frequency table, and apply the general summation formulas from scratch.

• Method 2: Directly use the algebraic properties of changes of origin and scale on the already known parameters: $\bar{x}' = a\bar{x} + b$ and $\sigma_{x'}^2 = a^2 \cdot \sigma_x^2$.

1. a) Apply Method 2 to numerically calculate $\bar{x}'$ and $\sigma_{x'}^2$.
2. b) Compare both methods and justify which one is more efficient when working with real databases containing thousands of records. Do both paths arrive at exactly the same result?
3. c) Explain in detail why the term $+5$ changes the value of the mean, but produces no change in the calculation of the marginal variance.

The National Statistics Institute (INE) studies, in a sample of 100 agricultural holdings, the crop area in hectares ($X$) and the annual volume of irrigation water used in thousands of cubic metres ($Y$):

$X \setminus Y$	4	8	12
2	30	10	0
5	10	25	5
10	0	5	15

1. a) Separately obtain the marginal distributions of $X$ and $Y$, and calculate for each of them its mean and its marginal variance: $\bar{x}$, $\sigma_x^2$, $\bar{y}$, and $\sigma_y^2$.
2. b) A preliminary technical report concludes: "Since the variance of irrigation water ($\sigma_y^2$) is numerically greater than the variance of area ($\sigma_x^2$), the holdings are more heterogeneous and dispersed in water consumption than in land size". Justify with reasons why this direct comparison between variances is not statistically valid.
3. c) Calculate the coefficient of variation for each variable ($CV = \frac{\sigma}{\bar{x}}$) and determine, based on this dimensionless indicator, which of the two variables actually exhibits greater relative dispersion.