X and Y are two independent random variables with variances 1 and 2, respectively. Let Z = X - Y. The variance of Z is (Answer up to the nearest integer)Correct answer is '3'. Can you explain this answer?

Question

Answer

Variance of Z = X - Y

To find the variance of Z, we need to determine the variance of the difference between X and Y. Since X and Y are independent random variables, their variances can be summed to find the variance of their difference.

Step 1: Find the variance of X
Given that the variance of X is 1, we can express it as Var(X) = 1.

Step 2: Find the variance of Y
Given that the variance of Y is 2, we can express it as Var(Y) = 2.

Step 3: Find the variance of Z
The variance of Z can be calculated using the formula: Var(Z) = Var(X) + Var(Y) - 2 * Cov(X, Y), where Cov(X, Y) represents the covariance between X and Y.

Since X and Y are independent random variables, their covariance is 0. This is because the covariance measures the linear relationship between two variables, and independence implies that there is no linear relationship between X and Y.

Therefore, the covariance term becomes 0, and the variance of Z simplifies to: Var(Z) = Var(X) + Var(Y) = 1 + 2 = 3.

Conclusion
The variance of Z, which is the difference between X and Y, is 3. This result is obtained by summing the variances of X and Y, since they are independent random variables. The covariance term is 0 due to the independence, simplifying the calculation.

Answer

Given: X and Y are two independent random variables with variances 1 and 2, respectively.

i.e. var(x) 1 & var(y) = 2

Let Z = x - y

var(z) = var(x - y)

= var(x) + var(y); since if x and y are two independent random variables, then

var(ax ± by) = a² var(x) + b² var(y)

var(z) = 1 + 2 = 3

var(z) = 3

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