Let $X, Y \in \text{Random}(\mathbb{R})$ each be a
D3161: Random real number such that
Let $\alpha, \beta, \gamma, \delta \in \mathbb{R}$ each be a
D993: Real number such that
| (i) |
\begin{equation}
\alpha
\neq 0
\neq \gamma
\end{equation}
|
Since $X$ and $Y$ are independent, then $\mathbb{P}(X \in E, Y \in F) = \mathbb{P}(X \in E) \mathbb{P}(Y \in F)$ for every borel set $E, F \subseteq \mathbb{R}$. Thus, if $E, F \subseteq \mathbb{R}$ are borel sets, then so are $\frac{E - \beta}{\alpha}$ and $\frac{F - \delta}{\gamma}$, whence
\begin{equation}
\begin{split}
\mathbb{P}(\alpha X + \beta \in E, \gamma Y + \delta \in E)
& = \mathbb{P} \left( X \in \frac{E - \beta}{\alpha}, Y \in \frac{F - \delta}{\gamma} \right) \\
& = \mathbb{P} \left( X \in \frac{E - \beta}{\alpha} \right) \mathbb{P} \left(Y \in \frac{F - \delta}{\gamma} \right) \\
& = \mathbb{P}(\alpha X + \beta \in E) \mathbb{P}(\gamma Y + \delta \in E)
\end{split}
\end{equation}
$\square$
