Expectation of Linear Transformation of Random Variable/Discrete

Theorem

Let $X$ be a discrete random variable.

Let $a, b$ be real numbers.

Then we have:

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

where $\expect X$ denotes the expectation of $X$.

Proof

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

\(\ds \)

\(=\)

\(\ds a \expect X + b\)

$\blacksquare$