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Combining two cumulative/differential distributions

Hi Team,

Professor Dr Kannan (from IIT Madras) teaching Engineering statistics using JMP has a small question. He has two cumulative/differential distributions which he want to merge into a single distribution at any chosen cut off point.

Can this be done with JMP? if so please help.

Thanks

Murali

Re: Combining two cumulative/differential distributions

What's the relationship between the two distributions? Mixture? Convolution?

Re: Combining two cumulative/differential distributions

Hi Don,

This is what I could understand from his conversation.

He has two probability distributions (continuous probability density functions) both going from '0' to infinity. Now, he wants to use one distribution from '0' to 'a' and the other from 'a' to infinity. Which is the most suitable cut off point for a?

There should not be continuity issues at point a.

Can JMP merge these two distributions because the new area under the curve should become unity as well ( a necessary condition for a probability distribution)?

Obviously two distributions may not be just joined together because the area is not equal to 1.

So, one may have to take cumulative distribution data for both distributions and see near equality of the cumulative distribution at cut off point a. Then a new cumulative distribution involving these two has to be generated and then differentiating the resulting composite cumulative distribution will give the new probability density function satisfying area under the curve equal to 1.

I hope this would help

Murali

Regards

Re: Combining two cumulative/differential distributions

If I understand correctly, I think that you have got all what you need, except for one piece. You want to find the intersections of the two cumulative distributions, if there is an intersection. The x-coordinate of the intersection is "a". How to get it? Nonlinear platform may help. Check out the root.jmp that I attached. It illustrates how to find the intersection of two straight lines.

I don't think that you need to differentiate on the merged cumulative distribution, because the tangents do not change at all on the merged cumulative distribution. So just grab the original density function and merge them at cutoff point.

When I mention "density function", it is interesting that you use the term "differential distribution". I think that they are the same thing. But the latter is only used in specific scientific domain, according to Google results that I searched. Is that the case? What is the user's specialty?

Re: Combining two cumulative/differential distributions

I was think more simplistically than @peng_liu. Could you merge them based on quantile? For example, find the quantile for distribution 1 at a, and use that to find the starting value for distribution 2.

Re: Combining two cumulative/differential distributions

Agree, @Don . That is another way. That is indeed what step-stress does. @Muralidhara here are some pictures to depict the differences. And you need to ask the user about the meaning of "merge".

First is a "quantile" approach, and one stitch the two highlighted pieces, but the part from B must be shifted to the left.

This is also a "quantile" approach, and one must shift the part from A to the right.

This is the "intersection" approach, and stitch the two highlighted pieces. No parts to shift.

This is another "intersection" approach, two highlighted parts are stitched. One thing important that I mentioned in the earlier post is that the intersection may not exist. Or trivially useless, e.g. cross at the origin.