Exercises: Conditional distributions

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

$X \setminus Y$	$1$	$2$	$3$
$10$	$2$	$5$	$1$
$20$	$0$	$4$	$3$
$30$	$4$	$1$	$0$

Write down the frequency table of the distribution of the variable $Y$ conditional on $X = 10$ and calculate the size of this subpopulation.

From the joint distribution of the variables $X$ and $Y$ given in the table:

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

Calculate the arithmetic mean of the distribution of $X$ conditional on $Y = 5$, that is, $\bar{x}_{|y=5}$.

In a study on two variables $X$ and $Y$, the distribution of $Y$ conditional on a specific value $x_i$ has been isolated, obtaining the following absolute frequencies:

$Y | X = x_i$	Absolute frequency ($f_{j|i}$)
$100$	$4$
$200$	$12$
$300$	$4$

Calculate the conditional relative frequencies $h_{j|i}$ for each value of $Y$ and check that they add up to $1$.

Look at the following contingency table:

$X \setminus Y$	$10$	$20$	$30$	$40$
$1$	$1$	$8$	$2$	$0$
$2$	$5$	$3$	$9$	$1$
$3$	$2$	$2$	$4$	$7$

Determine the mode of the distribution of $Y$ conditional on $X = 2$ and the mode of the distribution of $X$ conditional on $Y = 30$.

The conditional distribution of a variable $Y$ for a particular group $X = a$ has been extracted, resulting in the following data:

The group size is $N_{|x=a} = 20$.

The sum of the values is $\sum f_{j|a} \cdot y_j = 160$.

The sum of the squares is $\sum f_{j|a} \cdot y_j^2 = 1400$.

Calculate the conditional variance $\sigma^2_{Y|X=a}$.

The number of spelling mistakes ($Y$) in a dictation exercise is analysed according to the year group ($X$). The two conditional distributions for $X=3$ and $X=4$ are extracted:

Distribution $Y | X=3$:

Values $y_j$: $0, 1, 2, 3$

Frequencies: $2, 5, 2, 1$

Distribution $Y | X=4$:

Values $y_j$: $0, 1, 2, 3$

Frequencies: $5, 3, 2, 0$

Calculate the mean number of mistakes in each year group ($\bar{y}_{|x=3}$ and $\bar{y}_{|x=4}$) and indicate in which year group, on average, fewer mistakes are made.

Given the distribution of the variable $X$ conditional on $Y = 10$:

$X | Y = 10$	$2$	$4$	$6$
$f_{i|10}$	$3$	$5$	$2$

Given that the conditional mean is $\bar{x}_{|y=10} = 3,8$, calculate the conditional standard deviation $\sigma_{X|Y=10}$.

In a two-way table, the following conditional relative frequencies are given for the variable $Y$ with respect to two values of $X$:

For $X = 1$: $h_{1|1} = 0,2$ ; $h_{2|1} = 0,8$

For $X = 2$: $h_{1|2} = 0,5$ ; $h_{2|2} = 0,5$

Explain whether the variables $X$ and $Y$ are statistically independent or dependent based exclusively on these data.

In a survey on digital habits conducted at a high school, the relationship between school year ($X$) and weekly hours spent playing video games ($Y$) is analyzed. The data from $100$ students are gathered in the following table:

Year ($X$) \ Hours ($Y$)	$5\text{ h}$	$10\text{ h}$	$15\text{ h}$
4.º de ESO ($X=0$)	$10$	$25$	$15$
1.º de Bachillerato ($X=1$)	$5$	$20$	$25$

Lucía wants to calculate the mean weekly hours conditional on being in 1.º de Bachillerato (that is, $\bar{y}_{|X=1}$) and carries out the following calculation in her notebook:

$$\bar{y}_{|X=1} = \frac{5 \cdot 5 + 10 \cdot 20 + 15 \cdot 25}{100} = \frac{25 + 200 + 375}{100} = \frac{600}{100} = 6\text{ hours}$$

Lucía concludes that 1.º de Bachillerato students play on average $6$ hours a week.

1. a) Without performing complex calculations, explain why a result of $6$ hours is inconsistent solely by looking at the values taken by $Y$ ($5, 10$, and $15$) and the frequencies in the 1.º de Bachillerato row.
2. b) Identify the conceptual mistake Lucía made in the denominator of her formula when calculating a conditional distribution.
3. c) Correct the procedure and find the actual value of the conditional mean $\bar{y}_{|X=1}$.

From a sample of $200$ people taken from the Survey on Equipment and Use of Information and Communication Technologies in Households (INE), the number of online purchases made in the last month ($Y$: $0, 2$, or $5$ purchases) is analyzed according to age group ($X$: young group aged $18\text{ to }35$ years [$X=1$] and senior group over $65$ years [$X=2$]):

• Young group ($X=1$, $100$ people): $10$ make $0$ purchases, $40$ make $2$ purchases, and $50$ make $5$ purchases.

• Senior group ($X=2$, $100$ people): $70$ make $0$ purchases, $25$ make $2$ purchases, and $5$ make $5$ purchases.

1. a) Estimate: Before performing any numerical calculations, observe the frequency distribution of each subgroup. Estimate approximately between which values the mean number of purchases for each group should fall (for instance, will it be below or above $2$?) and argue in which of the two groups the data are visibly more concentrated around zero.
2. b) Check: Calculate the mean number of purchases conditional on being young ($\bar{y}_{|X=1}$) and the mean conditional on being a senior ($\bar{y}_{|X=2}$). Check whether the obtained values confirm your previous estimate.
3. c) In light of both conditional distributions, explain whether the number of online purchases made is independent of age in this sample.

In a nutrition study, the carbohydrate content in grams ($X$) and protein content in grams ($Y$) of several energy bars are related. The results are grouped in the following table:

$X \setminus Y$	$5$	$10$	$15$
$20$	$8$	$12$	$0$
$30$	$2$	$15$	$5$
$40$	$0$	$4$	$14$

Calculate the mean and standard deviation of the protein content for those bars that have exactly $30\text{ g}$ of carbohydrates (distribution $Y|X=30$).

The traffic authority has collected data on drivers' age ($X$, in years) and the number of fines received in the last year ($Y$):

$X \setminus Y$	$0$	$1$	$2$
$[18, 30)$	$15$	$25$	$10$
$[30, 50)$	$40$	$15$	$5$
$[50, 70)$	$35$	$5$	$0$

Using the class midpoint for the variable $X$, calculate the mean age of drivers who have received exactly $1$ fine in the last year ($\bar{x}_{|y=1}$).

On a farm, the amount of fertiliser applied per hectare ($X$, in tens of kg) and the yield obtained ($Y$, in tonnes) are analysed.

$X \setminus Y$	$2$	$4$	$6$
$1$	$5$	$1$	$0$
$2$	$2$	$8$	$2$
$3$	$0$	$3$	$9$

Calculate the variance of the yield conditional on $20\text{ kg}$ of fertiliser having been applied ($\sigma^2_{Y|X=2}$).

A survey is carried out in a school asking about gender ($X$: $1=\text{Chico}$, $2=\text{Chica}$) and preferred social network ($Y$: $1=\text{TikTok}$, $2=\text{Instagram}$, $3=\text{BeReal}$).

$X \setminus Y$	$1$	$2$	$3$
$1$	$30$	$20$	$10$
$2$	$45$	$30$	$15$

Calculate the conditional relative frequency distributions $Y|X=1$ and $Y|X=2$. In light of the results, are gender and preferred social network independent in this sample?

In a study on video games, the number of hours played per week ($X$) is related to the level reached in the game ($Y$). For the group of players spending $10$ hours ($X=10$), the following incomplete distribution is given:

$Y | X = 10$	$5$	$15$	$25$
$f_{j|10}$	$4$	$k$	$2$

Given that the mean level reached by this group is $\bar{y}_{|x=10} = 13$, set up an equation and determine the value of the unknown absolute frequency $k$.

In a factory, two different machines ($X=1$ and $X=2$) produce the same type of component. The number of defects per batch ($Y$) is analysed.

For Machine 1 ($X=1$): $\bar{y}_{|x=1} = 2,5$ defects, $\sigma_{Y|X=1} = 0,5$ defects.

For Machine 2 ($X=2$): $\bar{y}_{|x=2} = 8,0$ defects, $\sigma_{Y|X=2} = 1,2$ defects.

Calculate the conditional coefficient of variation for each machine ($CV = \frac{\sigma}{|\bar{y}|}$) and indicate which machine has a more consistent production in terms of the number of defects.

The following bubble chart represents the joint distribution of two discrete variables $X$ and $Y$. The size of the bubble and the adjacent text indicate the joint absolute frequency $f_{ij}$ for each pair of values $(x_i, y_j)$.

Extract the frequency table of the distribution of $X$ conditional on $Y = 20$ and calculate its arithmetic mean $\bar{x}_{|y=20}$.

At a weather station, maximum wind speed ($X$, in km/h) and relative humidity ($Y$, in %) are recorded. The data are grouped into two-dimensional intervals:

$X \setminus Y$	$[40, 60)$	$[60, 80)$	$[80, 100)$
$[0, 20)$	$2$	$10$	$18$
$[20, 40)$	$5$	$15$	$6$
$[40, 60)$	$12$	$4$	$0$

Using the class midpoints, calculate the mean relative humidity on those days when the wind speed was between $40$ and $60\text{ km/h}$ ($\bar{y}_{|x \in [40, 60)}$).

The Transport Consortium analyzes user satisfaction levels with the intercity bus service ($Y$: $1 =\text{ Low}$, $2 =\text{ Medium}$, $3 =\text{ High}$) based on the type of transit pass used ($X$: $1 =\text{ Youth Pass}$, $2 =\text{ Single Ticket}$). The sample of $300$ passengers yields the following data:

Pass ($X$) \ Satisfaction ($Y$)	$1$ (Low)	$2$ (Medium)	$3$ (High)	Total
$X=1$ (Youth Pass)	$20$	$50$	$30$	$100$
$X=2$ (Single Ticket)	$40$	$100$	$60$	$200$
Total	$60$	$150$	$90$	$300$

To find out whether service satisfaction depends on the type of transit pass used, two technicians propose different strategies:

• Sofía's method: She calculates the mean satisfaction conditional on each row ($\bar{y}_{|X=1}$ and $\bar{y}_{|X=2}$) and compares them.

• Mateo's method: He calculates the conditional relative frequency distribution of each row (that is, the percentage of users with levels $1, 2$, and $3$ within each ticket type) and compares them with each other and with the marginal distribution of $Y$.

1. a) Apply Sofía's method by calculating both conditional means.
2. b) Apply Mateo's method by obtaining the percentages for each row.
3. c) Choose the method: Although in this case both methods lead to a compatible conclusion, justify which of the two methods is theoretically the only rigorous one to demonstrate statistical independence between variables, and what limitation the other method presents (consider whether two distributions with very different frequencies could happen to have the same mean by chance).

The coaching staff of a basketball team in the Liga LEB Oro records two variables across $50$ games: the number of three-pointers attempted by their starting point guard in a game ($X$, taking the values $2$ or $6$ attempts) and the number of three-pointers made ($Y$, taking the values $1$ or $3$ made shots). The contingency table of absolute frequencies is as follows:

$X \setminus Y$	$1$	$3$
$2$	$f_{11}$	$f_{12}$
$6$	$f_{21}$	$f_{22}$

The following statistical information about the player's performance is available:

1. He played exactly $20$ games in which he attempted $2$ three-pointers and $30$ games in which he attempted $6$ three-pointers.

2. The mean number of three-pointers made conditional on having attempted $2$ three-pointers was $\bar{y}_{|X=2} = 1,4$.

3. The mean number of three-pointers made conditional on having attempted $6$ three-pointers was $\bar{y}_{|X=6} = 2,2$.

1. a) Set up a system of equations for the first row and another for the second row, and calculate the value of the four unknown frequencies ($f_{11}, f_{12}, f_{21}$, and $f_{22}$).
2. b) Using exclusively the given conditional means and the sizes of each subgroup, calculate the overall mean of three-pointers made ($\bar{y}$) across all $50$ games without needing to sum cell by cell.

Given the following contingency table with two unknown absolute frequencies, $a$ and $b$:

$X \setminus Y$	$2$	$6$
$10$	$4$	$a$
$20$	$b$	$2$

It is known that the mean of $Y$ conditional on $X=10$ is $\bar{y}_{|x=10} = 5$, and that the mean of $X$ conditional on $Y=2$ is $\bar{x}_{|y=2} = 16$.

Set up a system of equations and determine the values of the frequencies $a$ and $b$.

In a hospital, a new treatment is being tested. Variable $X$ indicates the treatment ($0=\text{Estándar}$, $1=\text{Nuevo}$) and variable $Y$ indicates the outcome ($0=\text{No curado}$, $1=\text{Curado}$). The patients are divided into two risk groups, forming two conditional distributions:

Low Risk Group:

Standard Treatment: $10$ cured out of $50$ patients.

New Treatment: $2$ cured out of $10$ patients.

High Risk Group:

Standard Treatment: $6$ cured out of $10$ patients.

New Treatment: $25$ cured out of $50$ patients.

Calculate the recovery rate (which is equivalent to the mean $\bar{y}$) for each treatment within each risk group. Then, combine all patients and calculate the overall recovery rate for each treatment. Which treatment seems better in the separate groups and which seems better overall? (This phenomenon is known as Simpson's Paradox).

In a statistical study on variable $X$ (job category) and $Y$ (salary in thousands of euros), there are three categories ($X=1, 2, 3$). The following data are known for the conditional distributions $Y|X=x_i$:

Category 1: $N_1 = 10$, $\bar{y}_{|x=1} = 20$, $\sigma^2_{Y|X=1} = 4$

Category 2: $N_2 = 20$, $\bar{y}_{|x=2} = 30$, $\sigma^2_{Y|X=2} = 9$

Category 3: $N_3 = 10$, $\bar{y}_{|x=3} = 40$, $\sigma^2_{Y|X=3} = 4$

First, calculate the overall marginal mean $\bar{y}$. Next, calculate the marginal variance $\sigma_Y^2$ using the variance decomposition formula (or by calculating the total sum of squares from the given data).

A theoretical bivariate joint relative frequency distribution (probabilities) is given by the expression $h_{ij} = k \cdot (x_i + y_j)$, where $X$ takes the values $\{1, 2\}$ and $Y$ takes the values $\{1, 2, 3\}$.

Construct the two-way table in terms of $k$. Knowing that the sum of all relative frequencies must be $1$, find the value of $k$. Subsequently, calculate the mean of the conditional distribution $X|Y=2$.

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

A financial analyst studies the relationship between the risk level of an investment fund ($X$, rated from $1$ to $3$) and its annual return ($Y$, in %).

$X \setminus Y$	$2$	$6$	$10$
$1$	$10$	$10$	$0$
$2$	$5$	$20$	$5$
$3$	$0$	$5$	$15$

The analyst defines the "performance ratio" of a risk level as the quotient of its conditional mean return and its conditional standard deviation: $R_i = \frac{\bar{y}_{|x=i}}{\sigma_{Y|X=i}}$.

Calculate this ratio for risk levels $X=2$ and $X=3$. Which risk level offers a better ratio according to this criterion?

We want to reconstruct a $2 \times 2$ contingency table of absolute frequencies $f_{ij}$ for the variables $X \in \{0, 1\}$ and $Y \in \{0, 1\}$. The following information is available about the marginal and conditional distributions:

• Marginal of $X$: $N_{x=0} = 30$, $N_{x=1} = 20$.

• Marginal of $Y$: $N_{y=0} = 25$, $N_{y=1} = 25$.

• Conditional mean: $\bar{y}_{|x=0} = 0,3$ (remember that since $Y$ takes the values $0$ and $1$, the mean coincides with the proportion of ones).

Calculate the four joint frequencies $f_{11}, f_{12}, f_{21}, f_{22}$ and construct the complete two-way table.

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

In a marine biology study, the depth at which a species lives ($X$, in metres) is related to its length ($Y$, in cm). The data are grouped into intervals of different widths:

$X \setminus Y$	$[0, 10)$	$[10, 30)$	$[30, 70)$
$[0, 50)$	$24$	$10$	$2$
$[50, 150)$	$6$	$18$	$16$

Using the class marks, calculate the variance of the length for the specimens living in the deep zone ($X \in [50, 150)$), that is, $\sigma^2_{Y|X \in [50, 150)}$.

Demonstrate numerically a fundamental property of statistical independence. Given the following table where $X$ and $Y$ are independent (their joint relative frequencies are the product of the marginal ones):

$X \setminus Y$	$2$	$4$	Total
$1$	$10$	$30$	$40$
$2$	$15$	$45$	$60$
Total	$25$	$75$	$100$

Calculate the marginal mean $\bar{y}$ and the conditional means $\bar{y}_{|x=1}$ and $\bar{y}_{|x=2}$. Check that, since they are independent, the equality $\bar{y} = \bar{y}_{|x=1} = \bar{y}_{|x=2}$ holds.

The Health Service of an autonomous community analyzes the efficacy of two treatments ($A$ and $B$) against a seasonal respiratory condition. The variable $Y$ indicates the clinical outcome ($1 =\text{ Cured}$, $0 =\text{ Not cured}$). The study is stratified into two population groups by age (Young and Elderly), yielding the following results:

• Young patient group ($120$ people):

• With Treatment $A$: out of $100$ treated, $80$ are cured.

• With Treatment $B$: out of $20$ treated, $18$ are cured.

• Elderly patient group ($300$ people):

• With Treatment $A$: out of $100$ treated, $20$ are cured.

• With Treatment $B$: out of $200$ treated, $50$ are cured.

1. a) For the young group, calculate the mean cure rate conditional on treatment $A$ ($\bar{y}_{|\text{Young}, A}$) and on treatment $B$ ($\bar{y}_{|\text{Young}, B}$). Do the same for the elderly group. Which treatment achieves a higher success rate in each of the two groups separately?
2. b) Aggregate the data by summing the people from both age groups to construct the overall two-way table between Treatment ($A$ or $B$) and Outcome ($0$ or $1$). Calculate the overall mean cure rate for each treatment ($\bar{y}_{|A}$ and $\bar{y}_{|B}$).
3. c) Compare the results obtained in part (a) with those in part (b). What paradoxical phenomenon do you observe? Relying on the concept of conditional distribution and the sizes of the subgroups to which each treatment was administered, explain why this apparent contradiction occurs (Simpson's Paradox) and reason what grounded clinical recommendation should be made.

Based on a representative sample of $200$ people employed in the service sector in Spain, a sociological study analyzes the relationship between work arrangement ($X$: $1 =\text{ In-person}$, $2 =\text{ Remote work}$) and the weekly hours spent at home on family care and domestic chores ($Y$: $5, 15$, or $25\text{ hours}$). It is known that $120$ people work in person and $80$ work remotely.

In the in-person work group ($X=1$), the distribution of dedicated hours is as follows: $36$ people spend $5\text{ h}$, $60$ spend $15\text{ h}$, and $24$ spend $25\text{ h}$.

1. a) Model under the independence hypothesis: If work arrangement and household hours were statistically independent variables, how many people in the remote work group ($N_{|X=2} = 80$) should report exactly $5, 15$, and $25$ domestic hours? Justify how you obtain these expected frequencies from the relative frequencies of the in-person group.
2. b) Real data and calculation: The actual data collected in the survey for the remote work group ($X=2$) turned out to be: $8$ people spend $5\text{ h}$, $32$ people spend $15\text{ h}$, and $40$ people spend $25\text{ h}$. Calculate the mean and variance of the distribution of hours conditional on each arrangement: $(\bar{y}_{|X=1}, \sigma^2_{Y|X=1})$ and $(\bar{y}_{|X=2}, \sigma^2_{Y|X=2})$.
3. c) Sociological and statistical interpretation: In light of the results of part (a) and the parameters calculated in part (b), conclude with reasoning whether the variables are dependent or independent. Interpret what the change in the conditional distribution reveals about the impact of remote work on taking on domestic responsibilities.