Simple linear regression: If Y and X are both normal, what's the exact null distribution of the parameters? Suppose $Y \sim{N(a,b)}$, $X \sim{N(c,d)}$, and $Y$ is independent of $X$. After sampling 25 observations from both $Y$ and $X$, I run the following regression model: $Y=\beta_{0}+\beta_{1}X + \epsilon$. I wish to test the hypothesis $H_{0}: \beta_{0}=0$ against the alternative $H_{1}: \beta_{0}\neq 0$.
My question is, since the distributions of $Y$ and $X$ are known, is there an exact 'null distribution' for the parameter $\beta_{0}$? If so, what is the distribution? By null distribution, I mean the sampling distribution of $\beta_{0}$ under the null hypothesis.
If anyone knows the answer assuming the true correlation coefficient between $Y$ and $X$ is 0.1, rather than assuming independence, that would be a big help also. This is all for a simulation study I'm working on.
 A: Since you have specified that $X$ and $Y$ are independent, the conditional mean of $Y$ given $X$ is:
$$\mathbb{E}(Y|X) = \mathbb{E}(Y) = c,$$
which implies that:
$$\beta_0 = c \quad \quad \quad \beta_1 = 0 \quad \quad \quad \varepsilon \sim \text{N}(0, d).$$
In this case there is nothing to test --- your regression parameters are fully determined by the distributional assumptions you have made at the start of the question.
Remember that a regression model is a model designed to describe the conditional distribution of $Y$ given $X$.  If you assume independence of these variables then this pre-empts the entire modelling exercise.
A: In simple linear regression the computation of the estimate of $\beta_0$ is:
$$\hat\beta_0 = \frac {1}{n} S_y + \frac {1}{n} S_x \frac {n S_{xy} - S_x S_y}{ n S_{xx} - S_x S_x}$$
with $S_x = \sum x_i $, $S_y = \sum y_i $, $S_{xx} = \sum x_i x_i $, $S_{xy} = \sum x_i y_i $
You could say it will be a linear sum of the $y_i$
$$\hat\beta_0 = \frac {1} {n} \sum c_i y_i $$
with 
$$c_i =\left( 1 + \frac {n x_i - S_x}{n S_{xx} - S_x S_x}  \right) $$
This does not seem to follow an easy distribution (or at least not a typical well known distribution) for both random $x_i $ and $y_i$ you have:
$$\hat\beta_0 \sim N(\mu, \sigma^2)$$
where $\mu$ and $\sigma$ are random variables themselves depending on the distribution of $X$ as well. (if every $y_i$ has an identical distribution $N(a,b)$ then $\mu = a$, independent from the distribution of $X$) 
However if you condition on $x_i$ then $\hat\beta_0$ follows a regular normal distribution (note that the $y_i$ do not need to be distributed according to identical Normal distributions) . 
In testing you often do not know the variance of this normal distribution and you will estimate it based on the residuals. Then you will use the t-distribution.
