Variance of 3X minus 2Y for independent normals

Question

Take independent normal random variables, $X$ with mean $2$ and variance $4$ , and $Y$ with mean $1$ and variance $4$ . Put $Z=3X-2Y$ . What is $\operatorname{Var}(Z)$ ?

Short answer

$52$ . For independent variables, $\operatorname{Var}(aX+bY)=a^2\operatorname{Var}X+b^2\operatorname{Var}Y=9\cdot4+4\cdot4=52$ .

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Lessons behind this question

Continuous Random Variables and Probability Density Functions
Independence
Covariance and Correlation

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