7. X & Y are two independent random variables with variances 1 & 2 respectively. let z=x-y, 
    then variance of Z is
    a) 0    
   b) 1    
   c) 2   
   d) 3

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.

Explore how to calculate the variance of Z = X - Y using independent random variables X and Y with variances 1 and 2, respectively. Understand the formula Var(Z) = Var(X) + Var(Y) and its application. Suitable for advanced high school students and those interested in probability theory.

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