Question

If the correlation coefficient \(\rho\) of \(X\) and \(Y\) exists, show that \(-1 \leq \rho \leq 1\). Hint: Consider the discriminant of the nonnegative quadratic function $$h(v)=E\left\\{\left[\left(X-\mu_{1}\right)+v\left(Y-\mu_{2}\right)\right]^{2}\right\\}$$ where \(v\) is real and is not a function of \(X\) nor of \(Y\).

Short answer

The given quadratic function \(h(v)\) was expanded and written in quadratic form. Its discriminant \(D\) was determined as a relation between variance of \(X\), \(Y\) and their covariance. As discriminants are by nature non-negative, from \(D \geq 0\), inequality for \(\rho\) follows: -1 \leq \rho \leq 1. Hence it's demonstrated that correlation coefficient always lies between -1 and 1.

Step-by-step solution

Define the function

Firstly, look at the provided function \(h(v)=E\left\{\left[(X-\mu_{1})+v(Y-\mu_{2})\right]^{2}\right\}\). This function is an expected value of a squared expression.

Expand the function

Expand this squared term within the expected value operator to get \(h(v)=E[X^2-2X\mu_{1}+\mu_{1}^2+v^2Y^2+2vY\mu_{2}-2v\mu_{2}X+v^2\mu_{2}^2] = v^2E[Y^2]-2v\mu_{2}E[X]+v^2\mu_{2}^2+E[X^2]-2\mu_{1}E[X]+\mu_{1}^2\). Consequently, we have a quadratic function in terms of \(v\).

Find Discriminant

Given the quadratic function \(av^2+bv+c\) with \(a=E[Y^2]-\mu_{2}^2, b=-2\mu_{2}\mu_{1}+2\mu_{2}^2, c=\mu_{1}^2-2\mu_{1}^2+\mu_{2}^2+E[X^2]-\mu_{1}^2\), the discriminant \(D=b^2-4ac\) should be non-negative.

Substitute \(D\) with variables related to correlation coefficient

Upon simplifying \(a, b\), and \(c\) we observe that they can be substituted with variance of \(X\) and \(Y\) and their covariance. i.e., \(a=Var[Y], b=-2\mu_{1}\mu_{2}+2\mu_{2}^2, c=Var[X], Cov[X,Y]=\mu_{1}\mu_{2}-E[XY]\). So \(\rho(\text{correlation coefficient}) = \frac{Cov[X,Y]}{\sqrt{Var[X]Var[Y]}}\). Substitute these terms in \(D\) and simplify.

Make final conclusion

It is observed from earlier that \(-1 \leq \ ho \leq 1\) is a direct conclusion from \(D \geq 0\), which is true because \(D\) is the discriminant of the quadratic function and must be non-negative. Hence, we have shown that the correlation coefficient always lies between -1 and 1.

Key concepts

Statistical Correlation

Statistical correlation is a measure that describes the strength and direction of a relationship between two variables. It is represented by the correlation coefficient, commonly denoted as \(\rho\) or \(r\). The value of the correlation coefficient ranges from -1 to 1, with -1 indicating a perfect negative correlation, 1 indicating a perfect positive correlation, and 0 indicating no correlation at all.

Understanding the bounds of the correlation coefficient is crucial for interpreting the relationship between variables. For example, a coefficient close to 1 suggests that as one variable increases, the other variable tends to increase as well. Conversely, a coefficient close to -1 suggests that as one variable increases, the other tends to decrease. This concept is essential in various fields such as finance, psychology, and biology, where finding the relationship between variables can lead to valuable insights.

There are different methods to calculate the correlation coefficient, but the most common one is the Pearson correlation coefficient. It is computed as the covariance of the two variables divided by the product of their standard deviations. Analyzing the discriminant of a quadratic function can be one way to demonstrate that the correlation coefficient falls within its defined range, which was the focus of our exercise.

Quadratic Function

A quadratic function is a second-degree polynomial function of the form \(f(x) = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants and \(a \eq 0\). The graph of a quadratic function is a parabola, which can open upwards or downwards depending on the sign of \(a\).

In the context of our exercise, the quadratic function \(h(v)\) was in terms of variable \(v\) and involved expected values, which reflect a blend of probability and algebra. When investigating the correlation coefficient, we considered the discriminant of this quadratic function, which is given by \(b^2 - 4ac\). In order for a quadratic equation to have real solutions, the discriminant must be non-negative. This property was used to show the constraints of the correlation coefficient. Because the quadratic function in our problem is derived from an expected value, which is always non-negative, this guarantees that the discriminant is non-negative, implying that \(\rho\) is bound between -1 and 1.

Expected Value

The expected value, denoted by \(E[X]\), is a fundamental concept in probability and statistics, representing the long-term average of a random variable over many independent trials. It's essentially a weighted average of all possible values that a random variable can take on, with the weights being the probabilities of the respective outcomes.

For example, if we're considering a simple dice roll, the expected value of the outcome is 3.5, since that's the average of all possible rolls of a fair six-sided die. In more complex scenarios, the expected value might involve the average of outcomes based on a probability distribution, such as the expected profits from an investment or the expected number of occurrences of an event.

In the step-by-step solution for the correlation coefficient, the expected value played a central role. The function \(h(v)\) was an expected value of a squared expression, reflecting how we can use the expected value to explore properties of a distribution, in this case, to derive and validate the bounds of the correlation coefficient.