Two-Variable Distributions

Classify the following pairs of bivariate statistical variables according to the type of linear correlation (positive, negative or zero) expected between them:

• The weight of a vehicle and its fuel consumption per 100 km.

• The number of hours of direct sunlight and the amount of energy produced by a solar panel.

• An adult's shoe size and their monthly salary.

• The continuous usage time of a laptop and the remaining battery percentage.

Look at the following scatter diagrams relating different variables in three different experiments. Giving reasons, match each graph to one of the following linear correlation coefficients: $r = -0.92$, $r = 0.05$, $r = 0.88$.

In an electronics factory, the number of defects found in a batch ($X$) and the production shift ($Y$, where 1 is morning, 2 is afternoon and 3 is night) were measured over 50 days. The joint frequency table is as follows:

Calculate the marginal distribution of the variable $X$ (number of defects) and determine its arithmetic mean.

In an electronics factory, the number of defects found in a batch ($X$) and the production shift ($Y$, where 1 is morning, 2 is afternoon and 3 is night) were measured over 50 days. The joint frequency table is as follows:

Obtain the distribution of the variable $X$ conditional on production having taken place exclusively during the night shift ($Y=3$).

In a clinical trial, 100 patients were classified according to their age group ($X$, in years) and their blood cholesterol level ($Y$, in mg/dL), grouped into class intervals.

Calculate the mean and standard deviation of the cholesterol level exclusively for patients in the age group $[40, 50)$. Use class midpoints for your calculations.

In a survey of 200 university students, the relationship between the number of weekly hours spent playing sports ($X$) and the reported stress level ($Y$, measured from 1 to 3) was studied. Some data from the contingency table have been erased:

Complete the table knowing that $30\%$ of the total respondents spend 2 hours playing sports and have a stress level of 2. Once completed, mathematically justify whether the variables are statistically independent.

In a study, the variables $X$ and $Y$ take the values $X \in \{2, 4, 6\}$ and $Y \in \{4, 6, 8\}$. The marginal distributions are identical and uniform (absolute frequency of 10 for each individual value, with a sample size of $N=30$). Furthermore, it is known that the joint distribution is perfectly symmetric ($f_{ij} = f_{ji}$) and that the joint frequency for $X=2$ and $Y=4$ is 8, for $X=2$ and $Y=6$ is 2, and for $X=4$ and $Y=6$ is 2.

Calculate the remaining joint frequencies in the table and prove that the covariance of this distribution is $\sigma_{xy} = 1.6$.

In a study on the performance of 12 servers, the amount of allocated RAM in GB ($X$) and the number of requests processed per second ($Y$) were recorded. Given that $\sum_{i=1}^{12} x_i y_i = 4500$, the mean RAM is $\bar{x} = 16$ GB and the mean number of requests is $\bar{y} = 22$, calculate the covariance $\sigma_{xy}$.

In an agricultural cooperative, the amount of fertiliser applied in kg/ha ($X$) and the crop yield in metric tons/ha ($Y$) were recorded across 5 experimental plots:

• $X$: 10, 20, 30, 40, 50

• $Y$: 2.5, 3.8, 4.2, 5.5, 6.0

Calculate the marginal means, marginal variances and covariance of this bivariate distribution.

The relationship between the drying time of an industrial paint ($X$, in hours) and its abrasion resistance ($Y$, on an index from 0 to 100) was analysed in 4 trials, yielding the following pairs of values: $(2, 40)$, $(3, 55)$, $(5, k)$, $(6, 85)$.

If it is known that the marginal mean of the resistance is $\bar{y} = 62.5$, first determine the value of the parameter $k$. Next, calculate the covariance $\sigma_{xy}$ and the gradient of the regression line of $Y$ on $X$.

Let $(X, Y)$ be a two-dimensional distribution with covariance $\sigma_{xy} = 15$ and correlation coefficient $r = 0.6$. If a linear transformation is performed on the variables to adapt them to a new measurement scale, such that $U = 2X - 3$ and $V = -0.5Y + 1$, calculate the new covariance $\sigma_{uv}$ and the new correlation coefficient $r_{uv}$ between $U$ and $V$.

The covariance between the daily mean temperature in degrees Celsius ($X$) and ice cream sales in hundreds of pesos ($Y$) at an ice cream parlor during summer is $\sigma_{xy} = 18.5$. If the marginal standard deviations are $\sigma_x = 4.2$ and $\sigma_y = 5.1$, calculate the linear correlation coefficient $r$ and interpret the result.

The following scatter diagram relates the concentration of a chemical pollutant in a river ($X$, in ppm) to the mortality rate of a fish species ($Y$, in %). Observe the presence of an outlier clearly distinct from the rest of the scatter plot.

Explain, giving reasons, how removing this outlier would affect the covariance $\sigma_{xy}$, the gradient of the regression line of $Y$ on $X$, and the linear correlation coefficient $r$.

Consider a bivariate distribution of variables $(X, Y)$ with strictly positive marginal standard deviations ($\sigma_x > 0$ and $\sigma_y > 0$). Prove analytically that if an exact linear functional relationship exists between both variables of the form $Y = aX + b$ (with $a \neq 0$), then the linear correlation coefficient is $r = 1$ if $a > 0$, or $r = -1$ if $a < 0$. (Hint: use the properties of the mean and variance under linear transformations).

In an analysis of the relationship between monthly digital marketing spend ($X$, in thousands of pesos) and sales revenue ($Y$, in thousands of pesos), the two regression lines have been obtained. The regression line of $Y$ on $X$ is $y = 2.5x + 200$ and the regression line of $X$ on $Y$ is $x = 0.36y - 40$. Determine the mean point (center of gravity) $(\bar{x}, \bar{y})$ of this bivariate distribution.

The analysis of macroeconomic data from 20 countries shows that the mean Gross Domestic Product per capita ($X$, in thousands of pesos) is $\bar{x} = 500$, with a standard deviation of $\sigma_x = 160$. The mean energy consumption per capita ($Y$, in MWh) is $\bar{y} = 12$, with a standard deviation of $\sigma_y = 4$. If the linear correlation coefficient is $r = 0.85$, determine the equation of the regression line of $Y$ on $X$.

In a study on the relationship between annual rainfall ($X$, in mm) and wheat yield ($Y$, in metric tons per hectare), it is known that the regression line of $Y$ on $X$ is $y = 0.15x - 12$. If the mean rainfall is $\bar{x} = 400$ mm and the correlation coefficient is $r = 0.75$, find the equation of the regression line of $X$ on $Y$.

In a sociodemographic analysis, the two regression lines relating the birth rate ($X$, per thousand) and life expectancy ($Y$, in years) of a region have been obtained. The equations of the lines are:

$3x + y - 105 = 0$

$x + 2y - 160 = 0$

Identify, giving reasons, which is the regression line of $Y$ on $X$ and which is that of $X$ on $Y$. Subsequently, calculate the marginal means $\bar{x}$ and $\bar{y}$, and the linear correlation coefficient $r$.

Prove mathematically that if the equations of the regression lines of $Y$ on $X$ and $X$ on $Y$ are expressed in the form $y = mx + n$ and $x = m'y + n'$ respectively, then the linear correlation coefficient satisfies $r = \pm \sqrt{m \times m'}$. Furthermore, give reasons as to what determines the sign of $r$.

The human resources department of a software company has determined that the regression line relating years of experience ($X$) to monthly salary in pesos ($Y$) of its programmers is $y = 1\,700x + 24\,000$. Estimate the monthly salary of an employee who has 5 years of experience.

The following scatter diagram shows the relationship between the temperature applied to a metal alloy ($X$, in $^\circ\text{C}$) and its thermal expansion ($Y$, in mm). From the visual representation of the scatter plot and the fitted regression line, estimate the value of the y-intercept and the gradient of the line. Subsequently, predict the expansion for a temperature of $80^\circ\text{C}$.

An e-commerce platform has modelled the relationship between the number of daily visits to a product page ($X$) and the number of units sold ($Y$) using the regression line $y = 0.04x + 2.5$. The correlation coefficient is $r = 0.91$ and the range of visits in the studied sample is between 100 and 500.

1. a) Estimate the sales if a product receives 350 visits.
2. b) Estimate the sales if a product receives 2000 visits.
3. c) Discuss the reliability of both predictions based on the data provided.

The growth of a bacterial colony as a function of time ($X$, in hours) does not follow a linear model, but an exponential one of the form $Y = A \times e^{BX}$, where $Y$ is the number of bacteria (in thousands). To apply linear regression techniques, the change of variable $Z = \ln Y$ is performed.

The transformed data $(X, Z)$ yield the following marginal and joint results for 5 observations: $\bar{x} = 4$, $\bar{z} = 2.1$, $\sigma_x^2 = 2.5$, $\sigma_{xz} = 1.2$.

Calculate the values of the parameters $A$ and $B$ of the original exponential model.

For a bivariate statistical distribution $(X, Y)$ of sample size $N = 10$, the following sums are known:

$\sum x_i = 50$, $\sum y_i = 70$, $\sum x_i^2 = 286$, $\sum y_i^2 = 540$, $\sum x_i y_i = 385$.

Calculate the coefficient of determination $R^2$ and the residual variance $\sigma_e^2$. Interpret the meaning of both values in the context of the goodness of fit of the linear model.