How to calculate a conditionally trancated PDF from an ordered set

[nikozm]

Hello,

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.

 

[mmwave]

Hello,

Thank you for your inquiry. I would like to provide some insights on evaluating the conditional PDF of x_j in the given condition.

Firstly, let's define some terms for better understanding. The given condition can be rewritten as x_1 ≥ x_2 ≥ ... ≥ x_L and x_L ≥ y, where x_1, x_2, ..., x_L are random variables and y is a fixed deterministic value.

Now, the conditional PDF of x_j (1 ≤ j ≤ L) given that x_L ≥ y can be evaluated by using the definition of conditional probability. It is defined as the probability of x_j occurring given that x_L ≥ y has already occurred. In other words, it is the probability of x_j occurring among the remaining x_j-1, x_j-2, ..., x_1 variables, given that x_L ≥ y.

From the probability perspective, there is a difference between the cases when x_L ≥ y and when x_L = y. This is because in the former case, there are more possible values for x_L, which can affect the probabilities of x_j. For example, if x_L is close to y, then the probabilities of x_j occurring will be higher compared to the case when x_L = y.

To evaluate the conditional PDF in this case, you can use the same formula as in the well-known case when x_L = y, but you will need to consider the additional possibility of x_L being greater than y. This can be achieved by integrating the conditional PDF of x_L over the range of values greater than y.

I hope this helps in your evaluation. If you need further assistance, please do not hesitate to ask.