# Can a linear regression output a Normal Distribution?

If I want to use a linear/multiple regression in order to predict a value, can the output of this regression be a normal distribution, with the mean value being the traditional output, but there being an additional output of the variance/standard deviation?

Ideally, what I'd like to do is to be able to know the confidence of my regression. I know this is something more akin to Bayesian inference, though it is a desirable property for my current application.

I guess this would make it no longer be a linear regression though, as the output is no longer a line, but a more complex curve. Is there a name for this kind of regression?

Is it maybe possible to do a second regression on the residuals to produce a function which encodes the distribution? Or is this possibly unnecessary?

• If you're predicting a "value" with a Bayesian approach, then the output certainly is a distribution. Under well-known assumptions for linear regression it will even be Normal. But what exactly does this have to do with "confidence" or that the "output is no longer a line" or that some "complex curve" is present? At least three different concepts seem intertwined here. Your edits to clarify what you are asking would be welcome.
– whuber
Mar 1, 2016 at 20:27
• Sounds like the OP desires a prediction interval? Mar 1, 2016 at 20:47
• @AndrewM Prediction intervals do appear to be what I'm looking for! Thank you! I found this slide deck which seems to matches the concept I was desiring exactly. I'll read more about this and come back if I have any more pertinent questions, as @ whuber pointed out. Since that was essentially what I was looking for, do you want to add that as an answer? Though I guess the question was a bit incorrect in that the interval is arguably not an 'output'. I can update the question to be more clear relative to the answer. Mar 1, 2016 at 21:39