I'm going to predict the random variable $Y_i$ by linear regression: $Y_i=\alpha + \beta X_i + \epsilon$.
Assuming that the dependent variable ($Y_i$) has a normal distribution with different means and the same variance, i.e., $Y_i \sim N(\mu_i, \sigma^2)$, can anybody explain the distribution of the estimated variable ($y$)?
Can I assume it(estimated variable by regression) as a normal distribution? Can I get the variance of this estimation?
UPDATE: ${X_i}$ is a sequence of observations, which I am assuming are drawn independently from a Normal latent variable, Y.