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I am trying to evaluate the following condition:

Let x_1 ≥ x_2 ≥ ... ≥ x_L and x_L ≥ y, where y is a fixed deterministic value.

What is the conditional PDF of x_j (1 ≤ j ≤ L), given that x_L ≥ y ??? (recall that i < L)

Note that the conditional PDF of x_j, for the case when x_1 ≥ x_2 ≥ ... ≥ x_L and x_L = y is well-known and it has been given in the textbook of H.-C. Yang and M.-S. Alouini, "Order Statistics in Wireless Communications," Eq. (3.12), page 42. (e.g., see the attachment file).

But i would like to evaluate the case when x_L ≥ y and not just x_L = y.

Is there any difference, from the probability perspective ?

Any help would be useful.

Thank you in advance.

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# How to calculate a conditionally trancated PDF from an ordered set

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