Expectation of Linear Transformation of Random Variable

Theorem

Let $X$ be a random variable.

Let $a, b$ be real numbers.

Let $\expect X$ denote the expectation of $X$.

Then we have:

$\expect {a X + b} = a \expect X + b$

if that expectation exists.

Proof

Discrete Random Variable

We have:

$\ds \expect {a X + b}$

$=$

$\ds \sum_{x \mathop \in \Img X} \paren {a x + b} \map \Pr {X = x}$

Expectation of Function of Discrete Random Variable

$\ds $

$=$

$\ds a \sum_{x \mathop \in \Img X} x \map \Pr {X = x} + b \sum_{x \mathop \in \Img X} \map \Pr {X = x}$

$\ds $

$=$

$\ds a \expect X + b \times 1$

Definition of Expectation of Discrete Random Variable, Definition of Probability Mass Function

$\ds $

$=$

$\ds a \expect X + b$

$\blacksquare$

Continuous Random Variable

Let $\map \supp X$ be the support of $X$.

Let $f_X : \map \supp X \to \R$ be the probability density function of $X$.

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