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.

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 tonnes/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.

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 $V$.

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 $r = \pm \sqrt{m \cdot m'}$. Furthermore, give reasons as to what determines the sign of $r$.

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 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 $\sigma_{xy} = 1,6$.