Question

# Suppose X and Y are independent continuous random variables uniformly distributed on the intervals 0 \leq x \leq 2 and 0 \leq y \leq 4, respectively. Compute the variance of \sqrt{2}X-\frac{1}{2}Y. Hint: First find the variance of X and the variance of Y.

Random variables
Suppose X and Y are independent continuous random variables uniformly distributed on the intervals $$0 \leq x \leq 2 and 0 \leq y \leq 4$$, respectively. Compute the variance of $$\sqrt{2}X-\frac{1}{2}Y$$. Hint: First find the variance of X and the variance of Y.

2021-05-19
Step 1
Given information
X and Y independent continuous random variables follows uniform distribution
$$0 \leq x \leq 2, 0 \leq y \leq 4$$
Step 2
$$Var(X)=\frac{(2-0)^{2}}{12}=\frac{4}{12}=\frac{1}{3}$$
$$Var(Y)=\frac{(4-0)^{2}}{12}=\frac{16}{12}=\frac{4}{3}$$
Variance of $$ax+by=a^{2}var(x)+b^{2}var(y)$$
$$Var(\sqrt{2}X-\frac{1}{2}Y)=(\sqrt{2})^{2}Var(X)+(-\frac{1}{2})^{2}Var(Y)=2\times \frac{1}{3}+\frac{1}{4}\times \frac{4}{3}=1$$