Problem 60 - An Introduction to Mathematical Statistics and Its Applications

Question

Prove that \(\rho(a+b X, c+d Y)=\rho(X, Y)\) for constants \(a, b, c\), and \(d\) where \(b\) and \(d\) are positive. Note that this result allows for a change of scale to one convenient for computation.

Step-by-Step Solution

Answer

The correlation coefficient \(\rho(a+bX, c+dY)\) is equal to \(\rho(X, Y)\). It confirms that correlation is scale and location invariant.

Step 1: Recall the definition of correlation

The correlation coefficient between two random variables, X and Y, is defined as \(\rho(X, Y) = \frac{E[(X - E[X])(Y - E[Y])]}{\sqrt{Var(X) \cdot Var(Y)}} \) where E is the expected value operator and Var denotes variance.

Step 2: Substituting given variables into the formula

We replace X with (a + bX) and Y with (c + dY) in the definition. The new expression is: \(\rho(a + bX, c+ dY) = \frac{E[((a + bX) - E[a + bX])((c + dY) - E[c+ dY])]}{\sqrt{Var(a + bX) \cdot Var(c + dY)}}\)

Step 3: Simplify the expectation terms

Using linearity of expectations, we simplify the numerator as E[(bX)(dY)] because constants are pulled out of the E term and E[constant] = constant, then the values of constants vanish. Similarly, apply variance properties to simplify the denominator, result will be \(\sqrt{(b^2 \cdot Var(X)) \cdot (d^2 \cdot Var(Y))}\). Therefore, \(\rho(a + bX, c+ dY) = \frac{E[(bX)(dY)]}{bd\sqrt{Var(X) \cdot Var(Y)}}\).

Step 4: Final Simplification

Now, taking \(bd\) outside from the expectation operator in the numerator, and cancelling it with the one in the denominator, we get: \(\rho(a + bX, c+ dY) = \frac{E[(X)(Y)]}{\sqrt{Var(X) \cdot Var(Y)}} = \rho(X, Y)\).

Key Concepts

Statistical Correlation

Statistical correlation is a measure that determines the strength and direction of a linear relationship between two quantitative variables. It is typically expressed by the correlation coefficient, denoted as \( \rho \) or \( r \) for sample correlation. When two variables move together, either in the same (\

Expected Value

In statistics, the expected value is a fundamental concept that provides a measure of the center of a random variable's distribution. Put simply, it's the average outcome one would expect if an experiment could be repeated infinitely many times. But to a student learning statistics, think don't have to be complex.

In mathematical terms, for a discrete random variable, it's calculated by summing the product of each outcome with its probability. For a continuous variable, it's determined through integration.When dealing with expected values, the linearity property is particularly valuable. It states that the expected value of a sum of random variables is equal to the sum of their expected values, regardless of whether the variables are dependent or independent.This property greatly simplifies the calculation of expected values in complex scenarios and is precisely what's used in the original exercise to simplify terms involving \( a + bX \) and \( c + dY \) where \( a \) and \( c \) are constants.

Variance Properties

Variance is a statistical measurement that describes the spread of a set of numbers. It calculates the average squared deviation from the mean and helps to understand how much variation there is from the average (expected value). Variance properties are pivotal for understanding how data dispersion influences other statistical measures, like the correlation coefficient.

Two key properties of variance important in the original problem are:

• The variance of a constant is zero, because a constant does not vary.
• The variance of a constant multiplied by a random variable is the constant squared times the variance of the variable, represented mathematically as \( Var(aX) = a^2 \times Var(X) \) for a constant \( a \) and random variable \( X \).

These properties help in simplifying the correlation formula when variables are scaled by constants, as seen in the exercise where scaling by \( b \) and \( d \) appropriately adjusts the variance terms.

Linearity of Expectations

The linearity of expectations is a principle in probability theory that states the expected value of a sum of random variables is equal to the sum of their expected values. This is true no matter whether the variables are independent or not. It can be represented as \( E[X + Y] = E[X] + E[Y] \).

This principle brings a level of simplicity to complex calculations involving random variables, as it allows us to handle each random variable separately despite potential interactions between them. In the context of the original problem, this property facilitated the simplification of the expectation term in the numerator of the correlation coefficient formula, leading to a clearer understanding of how the correlation between \( X \) and \( Y \) remains unchanged after the addition of constants and scaling.