Dear researchers,
I hope you can advise an answer or a reference for the following question:
For a normally distributed random process, what is the probability distribution of its upper and lower envelopes?
Best regards,
Mahdi
Dear Prof. Prykhodko,
Thanks very much for answer. I thought the Rayleigh distribution would apply only for the upper envelope, but what I understand from you that both envelopes will follow that distribution.
Best regards,
Dear Mahdi Al-Badrawi,
You are right, usually Relleya distribution is used for the upper envelope. But sometimes in the construction of the distribution of the envelope the minimum values with the opposite sign of lower envelope are used. Therefore, in my opinion, you can use Relleya distribution for the lower envelope by taking its minimum values with the opposite sign.
Dear Prof. Prykhodko,
This is what I thought as well. However, I have some colleagues who argue that the way I consider the envelopes is different from the way the Rayleigh distribution is defined. In my case, the envelopes are obtained by connecting (using Spilne interpolation) the maxima points (to create upper envelop) and minima points (to create lower envelop). Do you think, and based on aforementioned definition of the envelop, that the Rayleigh is still applied as a proper distribution?
Best regards,
Dear Mahdi Al-Badrawi,
If you want to build the distribution with negative values, it is clear that the Relleya distribution can not be used. In this case, in my opinion, you can use the Pearson family of distributions or the Johnson family of distributions.
Dear Prof. Prykhodko,
Thanks for your feedback, I'll dig in the direction of Pearson and Johnson family distributions.
Best regards,
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