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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Then:

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

\(=\)

\(\ds \int_{x \mathop \in \map \supp X} \paren {a x + b} \map {f_X} x \rd x\)

Expectation of Function of Continuous Random Variable

\(\ds \)

\(=\)

\(\ds a \int_{x \mathop \in \map \supp X} x \map {f_X} x \rd x + b \int_{x \mathop \in \map \supp X} \map {f_X} x \rd x\)

Linear Combination of Definite Integrals

\(\ds \)

\(=\)

\(\ds a \expect X + b \times 1\)

Definition of Expectation of Continuous Random Variable

\(\ds \)

\(=\)

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

$\blacksquare$