Exercises: Bivariate 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.

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 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 $\sigma_{xy}$.

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.

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 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.

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 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.

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.

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 $X$.

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 $80^\circ\text{C}$.

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 $V$.

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.

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.

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 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 $X$.

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.

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 $r$.

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.

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 $\sigma_{xy} = 1.6$.

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).