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 \(ax+by=a^{2}var(x)+b^{2}var(y)\)

\(Var(\sqrt{2}X-\frac{1}{2}Y)=(\sqrt{2})^{2}Var(X)+(-\frac{1}{2})^{2}Var(Y)=2\times \frac{1}{3}+\frac{1}{4}\times \frac{4}{3}=1\)

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 \(ax+by=a^{2}var(x)+b^{2}var(y)\)

\(Var(\sqrt{2}X-\frac{1}{2}Y)=(\sqrt{2})^{2}Var(X)+(-\frac{1}{2})^{2}Var(Y)=2\times \frac{1}{3}+\frac{1}{4}\times \frac{4}{3}=1\)