Understanding Independent Random Variables

In probability theory and statistics, the concept of independence between random variables is very important. Two random variables, let's call them X and Y, are considered independent if the outcome of one variable does not affect the outcome of the other. This means that knowing the value of X does not give you any information about the value of Y, and vice versa.

There are several equivalent ways to define independence between random variables X and Y. One common definition involves their probability distributions or cumulative distribution functions. However, a very useful and often tested condition for independence relates to their expected values.

Condition for Independence using Expected Values

A fundamental property of independent random variables X and Y is related to the expected value of their product, \(E(XY)\). The condition states that if X and Y are independent, then the expected value of their product is equal to the product of their individual expected values.

Mathematically, this condition is expressed as:

\[ E(XY) = E(X) E(Y) \]

This equation is a necessary condition for independence. In many contexts, especially when dealing with moments, this property is used directly to check for independence or as a consequence of variables being independent.

Analyzing the Given Options

Let's look at the provided options and see which one matches the condition for independent random variables X and Y:

• Option 1: \(E (XY) = XE (Y)\)
  This option includes the random variable X itself on the right side, not its expected value. \(E(Y)\) is a constant value, but X is a variable. This equation is generally not true for independent variables, or for any variables unless X is a constant.
• Option 2: \(E (XY) = E (X) E (Y)\)
  This equation states that the expected value of the product XY is equal to the product of the expected value of X and the expected value of Y. This is the correct and standard condition for the independence of two random variables X and Y in terms of expected values.
• Option 3: \(E (XY) = E (X) + E (Y)\)
  This equation relates the expected value of the product to the sum of expected values. The property of linearity of expectation states that \(E(X+Y) = E(X) + E(Y)\) for any random variables X and Y (whether independent or not). However, there is no general property that says \(E(XY)\) equals \(E(X) + E(Y)\) for independent variables.
• Option 4: \(E (XY) = YE (X)\)
  Similar to Option 1, this option includes the random variable Y itself on the right side. \(E(X)\) is a constant value, but Y is a variable. This equation is generally not true for independent variables, or for any variables unless Y is a constant.

Based on the analysis, the condition that defines two random variables X and Y as independent, among the given options involving expected values, is \(E (XY) = E (X) E (Y)\).

Revision Table: Key Conditions

Additional Information: Independence vs. Uncorrelated

It is important to note the relationship between independence and being uncorrelated. Two random variables X and Y are said to be uncorrelated if their covariance is zero, which is defined as \(Cov(X,Y) = E[(X - E(X))(Y - E(Y))]\). Expanding this, we get:

\[ Cov(X,Y) = E[XY - X E(Y) - Y E(X) + E(X)E(Y)] \]

Using the linearity of expectation:

\[ Cov(X,Y) = E(XY) - E[X E(Y)] - E[Y E(X)] + E[E(X)E(Y)] \]

Since \(E(X)\) and \(E(Y)\) are constants:

\[ Cov(X,Y) = E(XY) - E(X) E(Y) - E(Y) E(X) + E(X) E(Y) \]

\[ Cov(X,Y) = E(XY) - E(X)E(Y) \]

So, X and Y are uncorrelated if and only if \(E(XY) - E(X)E(Y) = 0\), which means \(E(XY) = E(X)E(Y)\).

Thus, the condition \(E(XY) = E(X)E(Y)\) is the definition of uncorrelated variables. However, for most practical purposes and distributions encountered in introductory statistics, if variables are independent, they are also uncorrelated. The reverse is not always true; uncorrelated variables are not necessarily independent, though there are exceptions like the case of jointly normally distributed variables where uncorrelated implies independent.

Therefore, the statement \(E (XY) = E (X) E (Y)\) is the condition that signifies that X and Y are uncorrelated, which is a consequence of, and often tested as, the condition for independence in many standard contexts.