Need to solve the following product: ∏ k = 1 N 1 { y k ≥ 1 } Is there a way to simplify it or write it in a closed-form expression?

Question

Need to solve the following product:      ∏          k      =      1        N              1              {              y        k            ≥      1      }      Is there a way to simplify it or write it in a closed-form expression?

Phillip Fletcher · Accepted Answer

This question often arises in statistics for the likelihood with an uniform density.Pending you know the value of each       y    i  , we can talk about order statistics :       y          (      1      )        =      min    i        y    i    ,  …  ,      y          (      N      )        =      max    i        y    i  Looking at your product, it is equal to 0 as soon as one       y    i   is smaller than 1. Thus, you want the smallest       y    i   to be bigger than 1. We have            ∏              k        =        1            N                      1                    {                  y          k                ≥        1        }              =                  1                    {                  y                      (            1            )                          ≥        1        }              .  It is easier to maximize/minimize the likelihood now.

Need to solve the following product: prod_(k=1)^N 1_({y_k >=1}) Is there a way to simplify it or write it in a closed-form expression?

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