Exercises: Marginal distributions

Given the following two-way table for the 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 table of joint frequencies:

$X \setminus Y$	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 by observing 5 individuals:

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

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

In the following two-way table, the marginal totals (the sums of the rows and columns) 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 complete 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

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

Observe the following table of joint frequencies:

$X \setminus Y$	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 ($Y$).

A biologist studies a species of amphibian, recording its mass $X$ (in pounds) and its length $Y$ (in inches). 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 dollars) 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 the points scored ($Y$) per game are analyzed. The joint 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 the points scored is $\bar{y} = 10$, find the value of the unknown frequency $f$.

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

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

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

The number of weekly study hours ($X$) and the score obtained on 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 scores ($Y$) and determine what percentage of students obtained a marginal score higher than 6.

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

$X \setminus Y$	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 analyze its relative dispersion.

The mean temperature $X$ (in °F) and the 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 $Y$. Check numerically that the sum of the marginal frequencies of $X$ coincides with the sum of the marginal frequencies of $Y$, and that both are equivalent to the total sample size $N$.

The amount of fertilizer $X$ (in lb/acre) and the crop yield $Y$ (in tons/acre) in different plots are studied:

$X \setminus Y$	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?

In a factory, the number of defects $X$ and the inspection time $Y$ (in minutes) of a batch of parts are analyzed. 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, the pressure $X$ (in atm) and the volume $Y$ (in gallons) 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 the income level $X$ (in thousands of dollars) and the 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 $Y$ in steel alloys are evaluated. 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	?	12	?	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 the Expected Return ($Y$) of a set of investment portfolios. The number inside each bubble indicates the joint frequency.

From the graph, 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 analyzed:

$X \setminus Y$	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 the 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 the new marginal mean and variance of $X$ be without needing to recalculate the whole table? Justify your answer based on the properties of the mean and variance.

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

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

Knowing that the final retail price ($X'$) is calculated by applying a fixed tax of $\$5$ 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 of the values individually.

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

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

Show 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 if the marginal distribution of $X$ is extracted first and the one-dimensional formula $\bar{x} = \frac{1}{N} \sum_{i} x_i \cdot n_{i \bullet}$ is applied. Show the step-by-step working for both methods.