Expectation of Linear Transformation of Random Variable

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Theorem

Let $X$ be a 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

Discrete Random Variable

Let $\map {\operatorname {supp} } X$ be the support of $X$.

Then:

\(\ds \expect {a X + b}\) \(=\) \(\ds \sum_{x \mathop \in \map {\operatorname {supp} } X} \paren {a x + b} \map \Pr {X = x}\) Expectation of Function of Discrete Random Variable
\(\ds \) \(=\) \(\ds a \sum_{x \mathop \in \map {\operatorname {supp} } X} x \map \Pr {X = x} + b \sum_{x \mathop \in \map {\operatorname {supp} } X} \map \Pr {X = x}\)
\(\ds \) \(=\) \(\ds a \expect X + b \times 1\) Definition of Expectation of Discrete Random Variable, Definition of Probability Mass Function
\(\ds \) \(=\) \(\ds a \expect X + b\)

$\blacksquare$


Continuous Random Variable

Let $\map {\operatorname {supp} } X$ be the support of $X$.

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

Then:

\(\ds \expect {a X + b}\) \(=\) \(\ds \int_{x \in \map {\operatorname {supp} } X} \paren {a x + b} \map {f_X} x \rd x\) Expectation of Function of Continuous Random Variable
\(\ds \) \(=\) \(\ds a \int_{x \in \map {\operatorname {supp} } X} x \map {f_X} x \rd x + b \int_{x \in \map {\operatorname {supp} } X} \map {f_X} x \rd x\) Linear Combination of Definite Integrals
\(\ds \) \(=\) \(\ds a \expect X + b \times 1\) Definition of Expectation of Continuous Random Variable
\(\ds \) \(=\) \(\ds a \expect X + b\)

$\blacksquare$