What is the correlation between RNP level and the CDI indication? I got a question regarding the connection between RNP (required navigational performance) accuracy and cdi sensitivity of the HSI.

According to the instrument flying handbook the standard for RNP on route would be 2Nm from the centerline.

So for me this means, that I will be at least 95% of my flight in that corridor (given, if raim  is available all the time)

So let's say I am off by 1 Nm enroute due to inaccuracy.

According to Rnp that would be still ok, however the question now is, what would my CDI show me? 

The gps obviously does not know that it is 1Nm off course (if so why would we have such a thing like rnp of 2Nm) so the CDI should be centered even though I am actually 1Nm off the correct course.

If this is true I theoretically could be e.g. 4 Nm off course even though I got only a 2 dot deflection on my HSI given that a full deflection would represent 5 Nm because it is not WAAS capable (2Nm rnp + 2 Nm for the two dots) 

Thanks for you help in advance...
The Performance Based Navigation concept allows to optimize the instrument procedure design with the aircraft navigation performance.
This concept is used en route, to reduce aircraft separation, and in terminal area to optimize arrival and departure procedures. The utmost development of Performance Based Navigation for approach, missed approach and departure is known under different names.
FAA initially referred to RNP SAAAR Operations, SAAAR standing for Special Aircraft and Aircrew Authorization Required.
ICAO now refers to RNP Operations with Authorization Required (RNP AR).

The estimation of the aircraft position is performed by the Flight Management System based on different sensors: Inertial (ADIRU), GPS, radio navigation (DME and VOR).
The FMS computes the best aircraft position estimate based on the following hierarchy of navigation modes:


GPS/inertial,
DME/DME/inertial,
VOR/DME/inertial, 
Inertial only.


The RNP capability depends on the availability of GPS/IRS integrity.

The GPS integrity mainly depends on the number of received satellites, the FMS will calculate the GPS estimated error as a function of the number of received satellites and which satellites among the most appropriate for the aircraft position.

The ADIRUs are capable to evaluate their own estimated error that they send to the FMS, similarly the FMS itself is able to calculate the ADIRUs  drift error based on previous ILS landing(if any)  as he compares the ADIRU position to the real aircraft position while on the ILS, This error is reduced to a mean average drift per hour of flight.

The FMS therefore is able to calculate an Estimated Position Error (EPE), thus allowing the CDU to display both the Required Precision, and the Estimated precision

With respect to the HSI, it displays a route as well the approach calculated by the FMS, within the estimated precision which normally is better than the required precision, the pilot is supposed to be on the route and not one dot  or 2 dots apart. When the CDI is centered you are within the Estimated Error;  However indeed if the CDI is not centered the error is augmented as you say in the question, but you should not add the RNP but the Estimated NP that shows on the CDU, because the FMS calculations are normally correct.

Of course if your equipments are faulty and mysteriously the multiple faults are not detected you may be neither within the RNP nor within the ENP.

Please refer to the following website for more details:

https://www.theairlinepilots.com/forumarchive/aviation-regulations/rnp-ar.pdf