Linear Transformation of Real-Valued Random Variable is Real-Valued Random Variable

Theorem

Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $X$ be a real-valued random variable on $\struct {\Omega, \Sigma, \Pr}$.

Let $a$ and $b$ be real numbers.

Then:

$a X + b$ is a real-valued random variable.

Proof

From the definition of a real-valued random variable, we have:

$X$ is $\Sigma$-measurable.

We want to verify that $a X + b : \Omega \to \R$ is a $\Sigma$-measurable function.

From Pointwise Scalar Multiple of Measurable Function is Measurable, we have:

$a X$ is $\Sigma$-measurable.

From Constant Function is Measurable, we have:

$x \mapsto b$ is $\Sigma$-measurable.

From Pointwise Sum of Measurable Functions is Measurable, we have:

$a X + b$ is $\Sigma$-measurable.

So:

$a X + b$ is a real-valued random variable.

$\blacksquare$