# Thread: 3 Norm Dists (correlated), prob co-occurence of 1% right tail

1. ## 3 Norm Dists (correlated), prob co-occurence of 1% right tail

Hi,
hoping someone can lead me in the right direction..

I'm looking at three normal distributions, specifically looking at the 1% extreme in each of these (say in one of the tails). So I have 3 extreme events (in 3 data sets), each defined as being the 1% right tail.

Now, if I assume the correlation between each of the data sets is 80%, can I work out the probability of all 3 of the 1% extreme events occuring?

So its obviously less than 1% (which would be the joint probability if the correlations were 100%). And would be 1%^3 if 0 correlation I believe.

I'd considered application of Bayes probability theorem, eg:
Prob(A and B) = Prob(B given A) * Prob(A)
[A being the first extreme, B the second]..

But I think I can only calculate Prob(B given A) based on historical co-occurence, of which the data series doesnt have enough data points to be accurate (eg have less than 200 data points).

Also considered the portfolio variance calculation, substituting in Probability (eg 1%, or Z(1%)=2.33) for standard deviation... but I don't know if thats a valid application of the formula (I know its normal application, just trying to find something that fits my purpose)..

Also looked at a probability tree, but goes back to not knowing the Prob(B given A), so can't progress down the branch.

Please point me in the right direction! Thanks

2. For the estimation, it should not be not hard to get the estimate of the mean vector and
the variance-covariance matrix. Once you get all these estimates, you can compute the
probability of the tail regions by (numerical) integration.

3. Originally Posted by BGM
For the estimation, it should not be not hard to get the estimate of the mean vector and
the variance-covariance matrix. Once you get all these estimates, you can compute the
probability of the tail regions by (numerical) integration.
Thanks for posting.

Ok, with an actual 3x3 variance-covariance matrix, and a mean vector containing the 3 means..
What do you mean by numerical integration? Can you explain that process or refer me to a link?

Thanks

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