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 the frequency table for 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 $1$.

Observe 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 specific group $X = a$ has been extracted, resulting in the following data:

The size of the group 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 is analyzed according to the grade level ($X$). 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 grade level ($\bar{y}_{|x=3}$ and $\bar{y}_{|x=4}$) and indicate in which grade level, on average, fewer mistakes are made.

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

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

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

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

In a nutrition study, the carbohydrate content in grams ($X$) and the 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 Department of Motor Vehicles has compiled data on the age of drivers ($X$, in years) and the number of traffic tickets 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 mark for the variable $X$, calculate the mean age of the drivers who have received exactly $1$ ticket in the last year ($\bar{x}_{|y=1}$).

On a farm, the amount of fertilizer applied per acre ($X$, in tens of pounds) and the yield obtained ($Y$, in tons) are analyzed.

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

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

A survey is carried out in a high 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 view 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 who dedicate $10$ hours ($X=10$), the following incomplete distribution is given:

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

Knowing 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 part. The number of defects per batch ($Y$) is analyzed.

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 homogeneous 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 $f_{ij}$ of each pair of values $(x_i, y_j)$.

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

In a weather station, the maximum wind speed ($X$, in mph) and the 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 marks, calculate the mean relative humidity on those days when the wind speed was between $40$ and $60\text{ mph}$ ($\bar{y}_{|x \in [40, 60)}$).

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 $b$.

A new treatment is tested in a hospital. The variable $X$ indicates the treatment ($0=\text{Estándar}$, $1=\text{Nuevo}$) and the 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 cure percentage (which is equivalent to the mean $\bar{y}$) for each treatment within each risk group. Then, group all the patients and calculate the overall cure percentage 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 the variable $X$ (job category) and $Y$ (salary in thousands of dollars), 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 two-dimensional distribution of joint relative frequencies (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 $X|Y=2$.

A financial analyst studies the relationship between the risk level of an investment fund ($X$, scored 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 between its conditional mean return and its conditional standard deviation: $R_i = \frac{\bar{y}_{|x=i}}{\sigma_{Y|X=i}}$.

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

We wish 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 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.

In a marine biology study, the depth at which a species lives ($X$, in feet) is related to its length ($Y$, in inches). 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, $\sigma^2_{Y|X \in [50, 150)}$.

Show 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, being independent, the equality $\bar{y} = \bar{y}_{|x=1} = \bar{y}_{|x=2}$ holds.