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Let's assume we have a simple linear regression y(x) = a + bx. Is there a way to obtain a probability density function of y for any given x? Would the concept of confidence regions be useful here in any way?

I'm not a big expert in statistics and your help will be very much appreciated

Thank you

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    $\begingroup$ Regression doesn't mean equality. Indeed, the linear fit can be extremely bad. So, you can't say much at all about one variable given a linear regression with a known variable when the correlation is arbitrary. $\endgroup$
    – Douglas Zare
    Commented Jan 21, 2010 at 14:45
  • $\begingroup$ what would change if correlation were 0.95? $\endgroup$
    – Alex
    Commented Jan 21, 2010 at 15:33
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    $\begingroup$ That could happen from one very low probability outlier which is far above the mean for both random variables, so you still couldn't say much of anything. You are pushing on a rope. You need some other tools than the linear regression or correlation to say anything meaningful about either the density function or confidence regions. $\endgroup$
    – Douglas Zare
    Commented Jan 21, 2010 at 17:04
  • $\begingroup$ This question can be answered, but you need to be more specific: a) what is the space on which x lies ? (i.e. is x a vector or a matrix) b) how many observations you have ? (what is the dimension of y) $\endgroup$
    – user3561
    Commented Jan 26, 2010 at 17:14

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You need to make your answer more precise. First, the regression should be
y = a + bx + e
where e has some distribution, let's say described by a density g(e)

Now, do you mean:

1) Obtaining the density f(y|x) theoretically?
This one is easy. By the transformation of densities
f(y|x) = 1/(a+bx)*g(y/(a+bx))

2) Estimating the density when you do not know the parameters of the regression model?
This one will depend on what you know and what you don't know, and what data you have at hand, and is a more complex issue, so you need to provide more information.

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