How do we modify the tails of norrmal/any kind of distribution by keeping other portion is same?

I want to generate random numbers several times with keeping pdf constant in the middle portion and changing the trail part pdf of both sides. The pdf curve can be symmetry or asymmetry, it doesn't matter.

Answer 1

[Paul]

Hard to say much, at least for me, without knowing more details about the starting PDF and how the tails are going to be modified.

One approximate approach that may be feasible is to form the CDF of the new PDF, either exactly or via numerical integration as the case may be, and then model the resulting CDF as a piecewise linear distribution

[BHUKYA VENKATESH]

I need to increse the probobility density of one side tail and decrease other side tail. So it becomes asymmetric.

or

I need to increse the probobility density of one side tail only.

[Paul]

Are you asking how to change the tails?

Or are you asking how to the define and sample from the new pdf based on the original pdf and a definiiton of how to change the tails?

[BHUKYA VENKATESH]

Yes I want to change the tail

Answer 2

[Jeff Miller]

As Paul says, the details are important because there are infinitely many ways to do what you are asking. One way to think about the problem is in terms of a mixture distribution. Your original distribution is a mixture of (say) 10% of scores from its low tail, 80% of scores from its middle, and 10% of its scores from its high tail. It would be possible to keep the same three ranges of scores but to rearrange the percentages from each range in any way you like as long as they add up to 100%.

[BHUKYA VENKATESH]

Can you give one example

[Jeff Miller]

Suppose your original distribution is uniform between 0-1. You could think of this as a mixture of 10% of scores in the range 0-0.1, 80% in the range 0.1-0.9, and 10% in the range 0.9-1. If you wanted to have more scores in the top tail, you could instead have 5%, 80%, and 15% in the three ranges.

You could use this approach with any continuous distribution (at least). After you shift probabilities around, the probability density function will have some discontinuities, but maybe that would be OK for you.

[BHUKYA VENKATESH]

I want keep the remaining curve se except tails

[Jeff Miller]

I guess I don't understand what you want, because I thought my example did that. The middle 80% from 0.1 to 0.9 stays the same; the probabilities only change for the scores in the top and bottom 10% tails.

[BHUKYA VENKATESH]

Is it possible for you to show that on PDF vs x plot? Can you do it ?

[Jeff Miller]

Sure, that is easy. For X from 0 to 0.1, the PDF is 0.5. For X from 0.1 to 0.9, the PDF is 1. For X from 0.9 to 1, the PDF is 1.5. Note that the total area under the PDF is 1.