How Are PDF and CDF of Order Statistics Related?

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The relationship between the Probability Density Function (PDF) and Cumulative Distribution Function (CDF) of order statistics follows the same principles as for any random variable. The CDF can be expressed as the integral of the PDF, specifically F_X(x) = ∫_{-∞}^{x} f_X(t) dt. This definition applies universally, confirming that the fundamental relationship between PDF and CDF holds true for order statistics as well. Understanding this relationship is crucial for analyzing the behavior of order statistics in probability theory. The discussion reinforces the general applicability of these definitions in statistical contexts.
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Hello,

Is the relation between the PDF and CDF of order statistics is as any PDF and CDF. i.e.:

F_X(x)=\int_{-\infty}^{x}f_X(t)\,dt

Regards
 
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Yes; that's a general definition.
 

Thread 'Hypothesis testing: Defining H0, HA hypotheses so that ( H_A)_A' makes sense'

The standard _A " operator" maps a Null Hypothesis Ho into a decision set { Do not reject:=1 and reject :=0}. In this sense ( HA)_A , makes no sense. Since H0, HA aren't exhaustive, can we find an alternative operator, _A' , so that ( H_A)_A' makes sense? Isn't Pearson Neyman related to this? Hope I'm making sense. Edit: I was motivated by a superficial similarity of the idea with double transposition of matrices M, with ## (M^{T})^{T}=M##, and just wanted to see if it made sense to talk...
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