So I have this simple linear regression model without intercept:
$Y_i = \beta*X_i + \epsilon_i$
where $\epsilon_i$ are independent and are $N(0,\sigma^2)$. What is the distribution of Y? I suspect it is also a normal variable but I don't know how to prove it. In my textbook it says that Y is a sum of a nonrandom quantity and a normal variable so it is distributed normally. Can someone explain this please?
Here $ X$ is a fixed variable and hence, $\beta X$ is a fixed quantity. $\varepsilon\sim N(0,\sigma^{2}) $ is the normal variable. As the distribution of the right hand side is normal, $Y$ is normal. Normality of $Y$ can be easily observed by considering the MGF of $Y$. By definition,
\begin{eqnarray*} M_{Y}(t)&=&M_{\beta X+\varepsilon }(t)=E(e^{t(\beta X + \varepsilon)})\\ &=&e^{t\beta X}E(e^{t\varepsilon})\\ &=&e^{t\beta X}\exp\{\dfrac{1}{2}t^{2}\sigma^{2}\}\quad \mbox{ since } \varepsilon\sim N(0,\sigma^2)\\ &=&\exp\{(\beta X)t +\frac{1}{2}t^{2}\sigma^{2}\} \end{eqnarray*} which is the MGF of Normal distribution with mean $\beta X$ and variance $\sigma^{2}$.