Show that if two random variables 𝑥 and 𝑦 are independent, then their covariance is zero

To show that the covariance between two independent random variables \( X \) and \( Y \) is zero, let's begin by reviewing the definition of covariance.

### Definition of Covariance:
The covariance between two random variables \( X \) and \( Y \) is given by:
\[
\text{Cov}(X, Y) = \mathbb{E}[(X - \mathbb{E}[X])(Y - \mathbb{E}[Y])]
\]
where \( \mathbb{E}[X] \) and \( \mathbb{E}[Y] \) are the expected values of \( X \) and \( Y \), respectively.

We can also rewrite the covariance formula in a more convenient form:
\[
\text{Cov}(X, Y) = \mathbb{E}[XY] - \mathbb{E}[X]\mathbb{E}[Y]
\]

### Independence of \( X \) and \( Y \):
When \( X \) and \( Y \) are independent, the expectation of their product \( \mathbb{E}[XY] \) is the product of their expectations. That is:
\[
\mathbb{E}[XY] = \mathbb{E}[X] \cdot \mathbb{E}[Y]
\]

### Proof:
Now, substitute this into the covariance formula:
\[
\text{Cov}(X, Y) = \mathbb{E}[XY] - \mathbb{E}[X]\mathbb{E}[Y]
\]

Since \( \mathbb{E}[XY] = \mathbb{E}[X] \cdot \mathbb{E}[Y] \), we have:
\[
\text{Cov}(X, Y) = \mathbb{E}[X]\mathbb{E}[Y] - \mathbb{E}[X]\mathbb{E}[Y] = 0
\]

Thus, if \( X \) and \( Y \) are independent, their covariance is zero.

### Conclusion:
The covariance between two independent random variables is zero.

Would you like more details or have any questions?

### Related Questions:
1. What is the meaning of zero covariance in practical terms?
2. Can two variables have zero covariance but not be independent?
3. How is covariance related to correlation?
4. What is the geometric interpretation of covariance?
5. What happens if \( X \) and \( Y \) are not independent?

### Tip:
Covariance measures the degree to which two variables change together. If it's zero, they don't vary together, but it doesn't always imply independence unless explicitly stated.

Show that if two random variables 𝑥 and 𝑦 are independent, then their covariance is zero.

Learn how to prove that the covariance of two independent random variables is zero. This problem covers key probability concepts, including the definition of covariance and the property of expectation for independent variables. Suitable for undergraduate students studying probability theory.

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