Problem 179 - An Introduction to Mathematical Statistics and Its Applications

Question

If two random variables \(X\) and \(Y\) are defined over a region in the \(X Y\)-plane that is not a rectangle (possibly infinite) with sides parallel to the coordinate axes, can \(X\) and \(Y\) be independent?

Step-by-Step Solution

Answer

Yes, two random variables X and Y can be independent irrespective of whether they are defined over a rectangular region or not.

Step 1 Identify the meaning of independent random variables

Two random variables X and Y are said to be independent if and only if the joint probability distribution function (PDF) of X and Y can be expressed as a product of their marginal PDFs, i.e., \(f_{XY}(x, y) = f_X(x) \cdot f_Y(y)\) or equivalently, if and only if the joint cumulative distribution function (CDF) of X and Y can be expressed as a product of their marginal CDFs, i.e., \(F_{XY}(x, y) = F_X(x) \cdot F_Y(y)\). This definition does not require the region over which X and Y are defined to be a rectangle with sides parallel to the axes.

Step 2 Determine the region relevance

The region where the random variables are defined does not impact their independence. The shape of the region of definition (i.e., whether it is a rectangle or not) does not affect the independence of X and Y. The independence is purely down to the relationship between the variables, and does not require for the variables to be defined over a rectangular region.

Step 3 Conclude from the information

Based on the above steps, it can be concluded that the region of definition of the random variables not being a rectangle does not negate the possibility of X and Y being independent. As long as the joint PDF (or CDF) of X and Y can be expressed as the product of their marginal PDFs (or CDFs), X and Y are independent.

Key Concepts

Joint Probability Distribution

When we talk about joint probability distribution, we're exploring how two random variables, say \(X\) and \(Y\), occur together. If you imagine variables \(X\) and \(Y\) as two events, the joint probability distribution offers insight into the probability of these two events happening at the same time.
This concept is fundamental when studying the relationships between multiple random variables. It is often denoted as \(f_{XY}(x, y)\) and gives us probabilities for all combinations of \(X\) and \(Y\).

• Example: Imagine rolling two dice. The joint probability distribution would give us the probability of each possible outcome of both dice rolled together, like getting a 3 on one die and a 4 on the other.
• Importantly, for two variables to be independent — which means they do not influence each other — the joint probability distribution can be expressed as the product of their individual (or marginal) probability distributions.

Marginal Probability Distribution

The marginal probability distribution is essentially focusing on one of the random variables from a joint distribution, effectively 'summing out' the other to understand just one. When we want to find the marginal probability of \(X\), we essentially aggregate the probabilities over all possible values of \(Y\), which gives us the standalone probability distribution of \(X\).
The term 'marginal' comes from how you often see it written in probability tables — in the margins!

• For example, consider again our dice. The marginal probability distribution for one die would give us the probability of landing a specific number on that one die, regardless of what happens with the second die.
• Mathematically, for continuous variables, you find the marginal probability of \(X\) by integrating the joint probability distribution over all \(y\) values: \(f_X(x) = \int f_{XY}(x, y) \; dy\).

Understanding the marginal distributions is crucial, especially when examining whether two variables are independent, as independence implies that each variable has no effect on the other.

Cumulative Distribution Function

The cumulative distribution function (CDF) provides a way to describe the probability distribution of a random variable by accumulating probabilities up to a certain value. It gives the probability that a random variable \(X\) will take a value less than or equal to \(x\).
This concept applies to single variables and to pairs (or more) in the case of joint cumulative distribution functions. For two random variables \(X\) and \(Y\), the joint CDF \(F_{XY}(x, y)\) gives the probability that \(X\) is less than or equal to \(x\), and \(Y\) is less than or equal to \(y\).

• Example: In stock market analysis, the joint CDF can help predict how two stocks might behave together over time.
• Just like with probability densities, for two random variables to be independent, their joint CDF must factor into the product of their marginal CDFs: \(F_{XY}(x, y) = F_X(x) \cdot F_Y(y)\).

This accumulated probability view is incredibly useful in both practical applications and theoretical probability, aiding in understanding how often events should occur up to a certain point.