Linear Combination of Real-Valued Random Variables is Real-Valued Random Variable

Theorem

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

Let $X$ and $Y$ be real-valued random variables on $\struct {\Omega, \Sigma, \Pr}$.

Let $\alpha$ and $\beta$ be real numbers.

Then:

$\alpha X + \beta Y$ is a real-valued random variable.

General Result

Let $n \in \N$.

Let $\sequence {X_i}_{i \mathop \in \N}$ be a sequence of real-valued random variables on $\struct {\Omega, \Sigma, \Pr}$.

Let $\sequence {\alpha_i}_{i \mathop \in \N}$ be a sequence of real numbers.

Then:

$\ds \sum_{i \mathop = 1}^n \alpha_i X_i$ is a real-valued random variable.

Proof

Since $X$ and $Y$ are real-valued random variables, we have:

$X$ and $Y$ are $\Sigma$-measurable functions.

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

$\alpha X$ and $\beta Y$ are $\Sigma$-measurable.

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

$\alpha X + \beta Y$ is $\Sigma$-measurable.

So:

$\alpha X + \beta Y$ is a real-valued random variable.

$\blacksquare$