Yes, if you use the covariance formula correctly, you will obtain the correct result when dealing with dependent random variables.

Understanding Covariance and Variance with Dependent Variables

The variance of the sum (or any linear combination) of dependent random variables includes not only the variances of the individual variables but also their covariances. The general formula for the variance of the sum of two random variables $X$ and $Y$ is:

$V(X + Y) = V(X) + V(Y) + 2 \cdot \text{Cov}(X, Y)$

For more than two variables, say $X_1, X_2, \ldots, X_n$, the variance of their sum is:

$V(X_1 + X_2 + \ldots + X_n) = \sum_{i=1}^n V(X_i) + 2 \sum_{1 \leq i < j \leq n} \text{Cov}(X_i, X_j)$

Applying the Covariance Formula to Your Scenario

In your case, if you are considering three flips (or outcomes of some random variable) denoted as $X_1, X_2,$ and $X_3$, the correct variance calculation of a linear combination like $3X_1 + 3X_2 + 3X_3 - 3$ (which simplifies to $3(X_1 + X_2 + X_3) - 3$) requires accounting for both variance and covariance:

1. Variance of Each Variable:

   • If each $X_i$ represents a flip modeled by $\text{Bin}(1, 0.5)$, then $V(X_i) = 0.25$.

2. Covariance Between Variables:

   • If $X_1, X_2,$ and $X_3$ are dependent, you need to include their covariances. The covariance between any two random variables $X_i$ and $X_j$ is denoted as $\text{Cov}(X_i, X_j)$.

The variance of $3(X_1 + X_2 + X_3) - 3$ simplifies to:

$V(3(X_1 + X_2 + X_3) - 3) = 9V(X_1 + X_2 + X_3)$

Given that:

$V(X_1 + X_2 + X_3) = V(X_1) + V(X_2) + V(X_3) + 2\text{Cov}(X_1, X_2) + 2\text{Cov}(X_1, X_3) + 2\text{Cov}(X_2, X_3)$

Plugging in the values:

$V(X_1 + X_2 + X_3) = 0.25 + 0.25 + 0.25 + 2\text{Cov}(X_1, X_2) + 2\text{Cov}(X_1, X_3) + 2\text{Cov}(X_2, X_3)$

$V(X_1 + X_2 + X_3) = 0.75 + 2\text{Cov}(X_1, X_2) + 2\text{Cov}(X_1, X_3) + 2\text{Cov}(X_2, X_3)$

Finally, multiplying by 9:

$V(3(X_1 + X_2 + X_3) - 3) = 9 \times (0.75 + 2\text{Cov}(X_1, X_2) + 2\text{Cov}(X_1, X_3) + 2\text{Cov}(X_2, X_3))$

Conclusion

Yes, by using the covariance formula and including the covariance terms between the dependent variables, you would correctly account for the dependencies between flips. This approach ensures you correctly compute the variance for any dependent random variables.

Questions for Further Exploration:

1. How do you calculate covariance between two dependent random variables?
2. Can you think of an example where ignoring covariance would significantly impact the result?
3. What methods can be used to determine if variables are independent or dependent?
4. How does the concept of correlation relate to covariance?
5. How might dependency between variables affect real-world scenarios, such as financial models or risk assessments?

Tip: Always verify whether variables are independent or dependent before performing variance calculations. This ensures that the results accurately reflect the underlying data structure.