Simple Linear Regression Question

Question

Let $Y_{i} = \beta_{0} + \beta_{1}X_{i} + \epsilon_{i}$ be a simple linear regression model with independent errors and iid normal distribution. If $X_{i}$ are fixed what is the distribution of $Y_{i}$ given $X_{i} = 10$?

I am preparing for a test with questions like these but I am realizing I am less up to date on these things than I thought. Could anyone explain the thought process used to approach this kind of question?

Answer 1

Let $\epsilon_i \sim N(0,\sigma^2)$. Then, we have:

$$Y_i \sim N(\beta_0 + \beta_1 X_i,\sigma^2)$$

Further clarification:

The above uses the following facts:

(a) Expectation is a linear operator,

(b) Variance of a constant is $0$,

(c) Covariance of a random variable with a constant is $0$ and finally,

(d) A linear combination of normals is also a normal.

Does that help?