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Subject: How to combine two probability density functions (PDFs)?
Date: Tue, 24 Aug 2010 19:29:23 +0000 (UTC)
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I am writing to seek help with modeling the probability of an event (age), which is bracketed by two ages, where each age is defined by a Gaussian probability density function (PDF).

To start simply, consider the case where an event is known to have occurred between two brackets. For example:
-The upper age bracket is described by a Gaussian age with a mean value of 5000 (yr) and a 1-sigma value of 200 (yr).
-The lower age bracket is also described by a Gaussian age, but with a mean value of 2000 (yr) and a 1-sigma value of 400 (yr).

My 1st question: how can I use the two age brackets to define a new PDF reflecting the probability of the event, which is known to have occurred between the two features? Note: my intuition is that the shape of the PDF should be a plateau between the brackets with Gaussian tails.

My 2nd question: how does the solution change where the distance between the end-member brackets decreases and the sigma values increase (e.g., when the tails of each age bracket overlap, especially given that the event defined by the old bracket has to have occurred prior to the event defined by the younger bracket)

Thanks for any help that you can provide!

Sincerely,
Ryan