Probabilistic output from OLS regression Given a data set x (1 dimension) and their output y, I fitted a OLS regression which gives me the slope and intercept least squares regression line.
How do I estimate the probabilistic output of the predicted value y being greater or equal to a certain value given a new x, i.e. P(y >= {certain value} | x = {a new x}) ?
Thanks!
 A: It's sounds like you are interested in a predictive posterior distribution, which is a Bayesian concept. The cdf of a particular conditional of that distribution function will give you an answer to your problem. The predictive posterior distribution starts by assuming you model coefficients are themselves a random variable, $B$, and calculating a distribution for $P(B=\beta|X,\theta)$ where $X$ represents both the independent and dependent variables and $\theta$ are hyperparameters, then integrating over the space of coefficients and hyperparameters:$$p(\hat{X}|X)=\int_\Omega \int_B p(\hat{X}|\beta)p(\beta|X,\theta)d\beta d\theta$$ You would then need to find the distribution of new dependent variables conditioned on the new regressor values, which involves another integration. Generally the coefficient distribution and subsequent integration is calculated using some kind of sampling. 
However, if you are fairly confident that your parameters are extremely close to the the generating parameters, then you could use a normal approximation based on the residual error:$$\begin{align}P(\hat{y}>c|\hat{x})&=P(\hat{r}>c-\beta^T\hat{x})\\&\approx\Phi\left(\frac{c-\beta^T\hat{x}}{\sigma_{\textrm{res}}}\right)\end{align}$$ where $\Phi$ is the cdf of the standard normal and $\sigma_{\textrm{res}}$ is the standard deviation of the model residuals.
