Problem 14 - A Modern Introduction to Probability and Statistics: Understanding Why and How

Question

Let \(X\) and \(Y\) be two random variables and let \(r, s, t\), and \(u\) be arbitrary real numbers. a. Derive from the definition that \(\operatorname{Cov}(X+s, Y+u)=\operatorname{Cov}(X, Y)\). b. Derive from the definition that \(\operatorname{Cov}(r X, t Y)=r t \operatorname{Cov}(X, Y)\). c. Combine parts a and \(\mathbf{b}\) to show \(\operatorname{Cov}(r X+s, t Y+u)=r t \operatorname{Cov}(X, Y)\).

Step-by-Step Solution

Answer

Covariance is linear: shifts don't alter it, and scaling adjusts by scalar multipliers.

Step 1: Understanding Covariance

The covariance of two random variables, \(X\) and \(Y\), is defined as \(\operatorname{Cov}(X, Y) = E[(X - E[X])(Y - E[Y])]\), where \(E\) is the expectation operator.

Step 2: Part a: Shifting Random Variables

To find \(\operatorname{Cov}(X+s, Y+u)\), we substitute into the definition: \(E[((X+s) - E[X+s])((Y+u) - E[Y+u])]\). Simplifying, \(E[X + s] = E[X] + s\) and \(E[Y + u] = E[Y] + u\). Substitute and simplify to find \(\operatorname{Cov}(X+s, Y+u) = E[(X - E[X])(Y - E[Y])]\), which gives us \(\operatorname{Cov}(X, Y)\), thus showing \(\operatorname{Cov}(X+s, Y+u) = \operatorname{Cov}(X, Y)\).

Step 3: Part b: Scaling Random Variables

To find \(\operatorname{Cov}(rX, tY)\), consider the expectation definition: \(E[((rX) - E[rX])((tY) - E[tY])]\). Here \(E[rX] = rE[X]\) and \(E[tY] = tE[Y]\). Plug these into the expectation: \(E[(r(X - E[X]))(t(Y - E[Y]))] = rtE[(X - E[X])(Y - E[Y])] = rt\operatorname{Cov}(X, Y)\).

Step 4: Part c: Combining Shifts and Scales

Combine the results from parts a and b. The covariance of the shifted and scaled variables \(\operatorname{Cov}(rX+s, tY+u)\) uses both principles. From part a, shifting does not affect covariance, and from part b, scaling multiplies the covariance by the product of scalars \(rt\). Therefore, \(\operatorname{Cov}(rX+s, tY+u) = rt\operatorname{Cov}(X, Y)\).

Key Concepts

Random Variables

Random variables are fundamental in probability and statistics. They are used to represent outcomes of random phenomena. A random variable, say \(X\), can take on different numeric values, each associated with a particular probability. For example, when you roll a standard six-sided die, the outcome \(X\) can be any integer from 1 to 6, each occurring with a probability of \(\frac{1}{6}\).

Random variables can be classified into two types: discrete and continuous. Discrete random variables have specific, countable values, like the die example. Continuous random variables, on the other hand, take on a range of values, for instance, a person's height or the time it takes to complete a race.

It's important to understand that random variables represent the uncertainty and variability in processes and are crucial for statistical analysis and probability theory.

Expectation Operator

The expectation operator, often denoted as \(E[X]\), is a critical concept in understanding random variables. It is essentially the "average" or "mean" value that a random variable takes on over numerous trials or occurrences. If you were to perform an experiment many times, the expectation would be the average of all the observed outcomes.

For a discrete random variable \(X\) with possible values \(x_i\) and corresponding probabilities \(P(x_i)\), the expectation is calculated by: \[ E[X] = \sum x_i \cdot P(x_i) \]

In contrast, for a continuous random variable, the expectation is calculated as: \[ E[X] = \int_{-\infty}^{\infty} x \cdot f(x) \, dx \]where \(f(x)\) is the probability density function.

Expectation helps in predicting future outcomes and is used to calculate other important statistical measures such as variance and covariance.

Shifting and Scaling in Probability

Shifting and scaling are important operations that affect how we interpret random variables. Shifting refers to adding or subtracting a constant value to/from a random variable. If \(X\) is a random variable and \(s\) is a constant, then \(X+s\) is the shifted random variable.

Here, the expectation becomes \(E[X+s] = E[X] + s\). It's important to notice that shifting does not change the variability or spread of the data; it only changes its location.

Scaling, on the other hand, refers to multiplying or dividing a random variable by a constant. For a constant \(r\), the scaled random variable becomes \(rX\), and the expectation is \(E[rX] = rE[X]\). Scaling changes the spread or dispersion of the data.

• Shifting causes a change in the mean of the random variable
• Scaling affects both the mean and the variance

These operations are crucial for adjusting data in statistical models and understanding transformations in probability distributions.

Covariance Properties

Covariance is a measure used to determine how much two random variables change together. Mathematically, for two random variables \(X\) and \(Y\), covariance is defined as:\[ \operatorname{Cov}(X, Y) = E[(X - E[X])(Y - E[Y])] \]

If the covariance is positive, it indicates that as one variable increases, the other tends to also increase. A negative covariance suggests that as one variable increases, the other tends to decrease.

Covariance is affected by shifting and scaling. When you add constants \(s\) and \(u\) to \(X\) and \(Y\) respectively, the covariance remains unchanged, meaning \(\operatorname{Cov}(X+s, Y+u) = \operatorname{Cov}(X, Y)\). When you scale \(X\) and \(Y\) by constants \(r\) and \(t\), the covariance becomes \(r t \operatorname{Cov}(X, Y)\). These properties are vital in statistical analysis, especially when normalizing or adjusting datasets.