Prove that for all random variable $X$, and any real numbers $a$, $b$, \[\mathrm{E}(aX + b) =

Question

Prove that for all random variable $X$, and any real numbers $a$, $b$, $$\mathrm{E}(aX + b) = a\mathrm{E}(X) + b$$ and $$\mathrm{Var}(aX + b) = a^2\mathrm{Var}(X)$$

Accepted Answer

Provided the required moments are finite, $\mathrm E(aX+b)=a\mathrm E(X)+b$ and $\operatorname{Var}(aX+b)=a^2\operatorname{Var}(X)$.

Properties of expectation and variance

Properties of expectation and variance

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