Question

. Determine the correlation coefficient of the random variables \(X\) and \(Y\) if \(\operatorname{var}(X)=4, \operatorname{var}(Y)=2\), and \(\operatorname{var}(X+2 Y)=15\)

Short answer

The correlation coefficient between the random variables X and Y is 0.75.

Step-by-step solution

Apply the formula for the variance of the sum of two random variables

The variance of the sum of two variables X and Y is given by \(\operatorname{var}(X+2 Y) = \operatorname{var}(X) + 4* \operatorname{var}(Y) + 2* \operatorname{cov}(X, 2Y)\). Given that \(\operatorname{var}(X)=4\), \(\operatorname{var}(Y)=2\), and \(\operatorname{var}(X+2 Y)=15\), we can plug these into the formula and solve for \(\operatorname{cov}(X, Y)\). Remembering that \(\operatorname{cov}(X, 2Y)=2*\operatorname{cov}(X,Y)\), we get: \(15 = 4 + 8 + 2*\operatorname{cov}(X,Y)\).

Compute the covariance

Solving for \(\operatorname{cov}(X,Y)\), we get: \(3 = 2*\operatorname{cov}(X,Y)\) or \(\operatorname{cov}(X,Y) = 1.5\).

Compute the correlation coefficient

The correlation coefficient \(\rho_{X,Y}\) is given by \(\rho_{X,Y} = \operatorname{cov}(X,Y) / (\sqrt{\operatorname{var}(X)} * \sqrt{\operatorname{var}(Y)})\). Plugging in the computed values, we get: \(\rho_{X,Y} = 1.5 /(2*1) = 0.75\).

Key concepts

Random Variables

Random variables are a fundamental concept in statistics and probability. They can be seen as variables whose possible values are numerical outcomes of random phenomena. For example, when rolling a die, the result is a random variable because it varies with each roll. Random variables are classified into two types: discrete and continuous.

Discrete random variables take on specific values, such as integer outcomes. In the die roll example, the random variable "X" might equal one of the numbers 1 through 6. Continuous random variables, however, can take on any value within a range or interval, such as the temperature on a given day, measured to infinite precision. Understanding these differences helps in analyzing and applying appropriate statistical methods.

• **Discrete Random Variables**: Numbers like 1, 2, 3.
• **Continuous Random Variables**: Any numeric value within a range.

Key to working with random variables is calculating measures such as mean, variance, and covariance, which together contribute to understanding variability and relationships within data.

Variance

Variance measures how spread out a set of values of a random variable are. It helps in understanding the variability or dispersion of the data. Mathematically, variance is the average of the squared differences from the mean.

This concept is crucial because it tells us how much individual numbers in a set deviate from the mean, helping to appreciate data's inconsistency.

Variance is denoted as \( \operatorname{var}(X) \) for a random variable \( X \). For example, if \( \operatorname{var}(X) = 4 \), it means that the numbers deviate from the mean by an average squared distance of 4. The square root of variance, called the standard deviation, provides another layer of understanding.

• **Low Variance**: Data points are close to the mean.
• **High Variance**: Data points are spread out over a wide range of values.

This calculation is essential for gauging uncertainty and variability within datasets, laying the groundwork for further statistical analyses and findings.

Covariance

Covariance is a measure that indicates the extent to which two random variables change together. It helps in understanding the direction of the linear relationship between them. If variables tend to show similar behavior, their covariance will be positive, whereas if they show opposite behavior, it will be negative.

The formula for covariance between two random variables \( X \) and \( Y \) is \( \operatorname{cov}(X,Y) \). In our exercise, after applying the variance formulas, we found \( \operatorname{cov}(X,Y) = 1.5 \). This illustrates a positive relationship, although not necessarily a strong one.

Understanding covariance is valuable, particularly when combined with variance, as it leads to the calculation of the correlation coefficient. However, beware, covariance by itself does not specify the strength of the linear relationship; this is where the correlation coefficient takes the stage.

Mathematical Statistics

Mathematical statistics is the application of mathematics to the field of statistics. It involves using mathematical theories and methods to collect, analyze, and interpret data. This field combines elements of probability theory with statistical analysis to develop models and derive insights from data.

Key elements include:

• **Probability Theory**: Foundation for describing random variables and understanding statistical relationships.
• **Inference**: Draw conclusions about a population based on a sample.
• **Hypothesis Testing**: Assesses assumptions with statistical evidence.

In the context of our exercise, mathematical statistics provides the methods needed to calculate the correlation coefficient for random variables \( X \) and \( Y \). This involves using known formulas and statistical properties to derive insights. Practical applications of mathematical statistics can be seen across various fields such as economics, biology, engineering, and social sciences, emphasizing its utility in solving real-world problems.