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NickB

Thoroughly confused about Normal Dist'n and Truncation

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At the moment I'm meant to be preparing a project for my degree & have managed to get my knickers into a bit of a twist over normal distribution functions... I'm ment to be predicting probabilities of an earthquake occuring, to do this you use a technique called PSHA. Essentially you find the probability of a random variable Y exceeding a given ground motion, y*, given distance r, event magnitude m. The probability of exceedance is:
E(y*) = ∑Ni=1 αi ∫ ∫ ƒ(m) ƒ(r,d) Ρ(Y>y* | m,r) dr dm
   
Dealing with ƒ(m) (Guttenburg-Richter [bound] usually), ƒ(r) and αi is easy enough, and you will deal with the integrations numerically (using ranges of R and M). What you are then left over with is the probability function, Ρ(Y>y* | m,r,d), here to find the probability you find the estimated ground motion parameter (say PGA), which typically has a form like:
log PGA = c1 + c2M + c3log(r) + c4r ± εσ   
Where c1...c4 are constants relating to such things as the rupture mechanism, σ is the expected scatter of the residules (ie standard deviation) and ε is a number of SD's. You can then take log(y*) and σ to find the normal variant & look up (/calculate) the probability from the normal tables (or actually, you are supposed to use a truncated distribution, often at about 6σ ) This is the bit where I start to get a bit confused however, according to my supervisor (who's not here for a few weeks) you should numerically integrate a truncated range of ε values (say -6 .. +6), find the normal probability of exceedence (eg ε == -1 → p = 84%) then multiply with the predicted log(PGA) from the ε value ... rince, repeat for the whole range, summing all the log(PGA) values together (at least, this is how I've understood what he's saying), and an important thing to do is to truncate the range. I'm not too happy about this because I can't find any references to doing this (papers, books etc), does this make (logical) sense to anyone? I do have the source code for some software that does PSHA, and although I'm assured that it does do this, I can not find any evidence of something like this happening, nor can I find any references to this in the documentation that the authors have written to accompany the software (they generally discuss all the theory that is implemented in the software). Basically, if that made any sense to anyone, could someone please clear up (explain) why the whole iterating ε thing makes sense. I'm getting rather confused & slighly perturbed by that. Also I think that there are two seperate trucations I can make; one on ε numerical integration and a different one for the normal distribution, chopping off the tails - have I got that right? Thanks for reading this somewhat long post, I hope that this isn't too taramount to asking for the solution to a homework problem...there is rather more to the whole thing, but I do seem to be rather hung up over this slight issue! edit: fix unexpected smiley that turned up... [edited by - NickB on May 17, 2004 7:25:09 PM]

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