Understanding Expectation of Product of Random Variables

The question asks under which condition the mathematical expectation of the product of two random variables equals the product of their individual expectations. This property is a fundamental concept in probability theory.

Mathematical Expectation Explained

The mathematical expectation, often denoted as $E[X]$ for a random variable $X$, represents the weighted average of all possible values that the random variable can take. It's calculated by summing the product of each possible value and its corresponding probability.

The Property: Expectation of a Product

The statement in question is: $E[XY] = E[X]E[Y]$ where $X$ and $Y$ are two random variables.

The Role of Independence

This property, $E[XY] = E[X]E[Y]$, holds true if and only if the random variables $X$ and $Y$ are independent.

Independence means that the occurrence of one event (or the value taken by one variable) does not affect the probability of the other event (or the value taken by the other variable).

Mathematically, if $X$ and $Y$ are independent, then for any measurable functions $g$ and $h$, it follows that: $E[g(X)h(Y)] = E[g(X)]E[h(Y)]$ Setting $g(X) = X$ and $h(Y) = Y$, we get the specific property mentioned in the question:

$E[XY] = E[X]E[Y]$

Why Other Options Are Incorrect

• Any two random variables: This is not always true. If $X$ and $Y$ are dependent, $E[XY]$ might not be equal to $E[X]E[Y]$. For example, if $Y=X$, then $E[XY] = E[X^2]$, which is generally not equal to $(E[X])^2$.
• If the covariance between the random variables is nonzero: Covariance ($Cov(X, Y)$) measures the degree of joint variability between two random variables. If $Cov(X, Y) \neq 0$, it implies that $X$ and $Y$ are dependent. In such cases, the property $E[XY] = E[X]E[Y]$ does not necessarily hold. In fact, if $X$ and $Y$ are dependent, $E[XY]$ is often different from $E[X]E[Y]$.
• If the variance of the random variables is equal: The equality of variances ($Var(X) = Var(Y)$) does not guarantee the independence of $X$ and $Y$, nor does it ensure that $E[XY] = E[X]E[Y]$.

Conclusion

The condition under which the mathematical expectation of the product of two random variables is equal to the product of their expectations is when the two random variables are independent.