G) For two random variables 𝑋 and 𝑌, 𝐸(𝑋𝑌) = 𝐸(𝑋)𝐸(𝑌) hold if 𝑋 and 𝑌 are _______

The statement $E(XY) = E(X)E(Y)$ holds if the random variables $X$ and $Y$ are independent.

Explanation:

Independence between two random variables means that the occurrence of one variable does not affect the occurrence of the other. Therefore, knowing the value of one random variable provides no information about the value of the other.

When two random variables are independent, their joint expectation can be expressed as the product of their individual expectations. This property is crucial in probability theory and statistics, as it allows us to simplify calculations involving multiple random variables.

To further illustrate, if $X$ and $Y$ are independent:

• The joint expectation $E(XY)$ can be decomposed based on the independence condition.
• This is a fundamental property used in various applications such as in analyzing random processes, conducting hypothesis tests, and estimating parameters.

In contrast, if $X$ and $Y$ are not independent, the relationship between their joint expectation and their individual expectations may not hold, and additional information about their joint distribution would be necessary to compute $E(XY)$.