transformation of probability distribution

May 2010

my question is about transfering a probability distribution from one parameter domain to another but the transfer function is not linear.

I hope I used the correct terminology above. What i want is the following. (also see the attached picture: transferFunc.png)

The relation between the position (x) and the measured quantity (y) at each position is the first half of a Gaussian function (shown in blue). I would term this my transfer function.

Let's assume that if I want to measure at the position x=10, I know that the positioning device is subject to noise and I model it with a normal distribution (shown in red). Thus I might measure at a slightly different position than assumed.

However, I want to describe this normal distribution in the domain of the measured quantity y. Can I compute the probability distribution in y (shown in green) for the Gaussian transfer function?

A linear relationship would be easy, right? Or is a transformation as how I describe it nonsense?

Thank you for your replies!