# variance of theoretical mean of y at given value of x in regression [duplicate]

Could someone tell me where I might find information on deriving the standard error used in confidence and prediction intervals of y at a given value of x on a regression line. I can't find anything on the internet. Thanks for any help.

• If it's Simple Linear Regression, then the expression can be derived easily by expanding the variance term into variance of both the parameters and the covariance between them.Also in S.E for Prediction Intervals,sigma square term is added. You may refer to this problem [link] (cheenta.com/isi-mstat-psb-2013-problem-4-linear-regression) – napoleon Jun 27 '20 at 4:31
• Thanks a ton, Napoleon. I was able to derive it like you said. – KWilliamson Jun 27 '20 at 22:34

## 1 Answer

In the regression model, knowing that Y is Normally distributed, one can use properties of the multivariate normal distribution to provide an answer.

I would recommend you look at Wikipedia's comment, especially for example, in the case of the bivariate normal distribution, and in particular, the conditional distribution of Y given X.

There is also an associated expression for the variance of the conditional distribution, which answers your question. See also, summary presented on Page 4 of this educational reference.

Note, for a prediction interval, one adds an added sigma square term.

• Thanks, AJKOER. To follow up on your last sentence, since we add the variances of the mean of y and the error of y, am I correct that they are assumed independent, since there is no covariance term in the formula? – KWilliamson Jun 27 '20 at 22:41