# Thread: Expectation of truncated normal X conditional on truncated normal Y

1. ## Expectation of truncated normal X conditional on truncated normal Y

I am trying to derive:

where , and are (doubly truncated) Gaussians with the same mean and different variance, and are the truncation points.

To start off, I wrote:

where

At this point I'm absolutely stuck. Is what I wrote correct? Is there any other way to derive the result more directly?

I've attached a Pdf as well for you to see the formulas more clearly, if needed.

Any help and advice would be GREATLY appreciated.

2. ## Re: Expectation of truncated normal X conditional on truncated normal Y

Several questions need to clarify:

1. is truncated normal on and is on ?

2. Are they independent? Or the untruncated X and Y are dependent, say bivariate normal? If they are independent, then the conditional expectation will be the same as the ordinary expectation.

3. They have the same mean, or the untruncated version of them having the same mean?

3. ## Re: Expectation of truncated normal X conditional on truncated normal Y

Hi BGM,

1. Yes, is truncated normal with truncation points , while is truncated normal with truncation points , where .

2. No, and are not independent. In fact, , where is also truncated normal with truncation points , and and are independent.

3. and have the same mean. More specifically: the truncated distribution of has mean zero. The truncated distribution of also has mean zero. Since and are independent, the density of can be reasonably approximated by a truncated Gaussian with mean equal to .

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