Mean of a linear combination of two normals

Question

Two normal random variables are given: $X$ has mean $2$ and variance $4$ , while $Y$ has mean $1$ and variance $4$ . Define $Z=3X-2Y$ . What is $\mathbb{E}[Z]$ ?

Short answer

$4$ . Expectation is linear regardless of dependence: $3\cdot2-2\cdot1=4$ .

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Lessons behind this question

Expectation, Mean, and Variance
Conditional Expectation and Variance Revisited
The Normal Distribution

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