To find the variance of $Z$ where $Z = X - Y$, we can use the property of the variance of the difference of two independent random variables.

Given:

• $X$ and $Y$ are independent random variables.
• Variance of $X$, $\text{Var}(X) = 1$.
• Variance of $Y$, $\text{Var}(Y) = 2$.

Since $Z = X - Y$, the variance of $Z$ is given by:

$\text{Var}(Z) = \text{Var}(X - Y)$

For independent random variables $X$ and $Y$, the variance of their difference is the sum of their variances:

$\text{Var}(Z) = \text{Var}(X) + \text{Var}(Y)$

Substituting the given variances:

$\text{Var}(Z) = 1 + 2 = 3$

Thus, the variance of $Z$ is $3$.

So, the correct answer is:

d) 3

Would you like more details or have any questions? Here are some related questions to expand your understanding:

1. How would the variance of $Z$ change if $X$ and $Y$ were not independent?
2. What is the variance of the sum $X + Y$ for the same independent variables $X$ and $Y$?
3. How do you calculate the standard deviation of $Z$?
4. If $X$ and $Y$ had means $\mu_X$ and $\mu_Y$, respectively, what would be the mean of $Z$?
5. How does covariance between $X$ and $Y$ affect the variance of $Z$ if $X$ and $Y$ were not independent?

Tip: When dealing with variances of sums or differences of independent random variables, remember that the variances add up, whether you are adding or subtracting the variables.