MHB Probability, Expected value, joint P.D.F. and order statistics

AI Thread Summary
The discussion focuses on understanding the derivation of a specific term in a mathematical example related to probability and order statistics. Participants are seeking clarification on several key concepts, including the meaning of a particular term, the derivation of a formula involving $n(n-1)(y-x)^{n-2}$, and proof of another statement. Additionally, there is a request for a computer program to verify results for Kth order statistics with sample sizes of n = 3 and n = 4. Contributors are encouraged to provide insights and relevant reading materials to address these queries. The thread emphasizes the need for clear explanations and practical applications in probability theory.
WMDhamnekar
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I want to know how did author derive the red underlined term in the below given Example?
Probability,Expectedvalue.png


Would any member of Math help board enlighten me in this regard?

Any math help will be accepted.
 
Last edited:
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Following reading material is necessary to answer my question. After reading the following material, you may answer my following questions and therby clear my doubts:
1) What is the meaning of 1 ?

2) How did author make 2 = $n(n-1)(y-x)^{n-2} ?$

3) How to prove 3 ?

Order Statistics:
1655698746147.png


Kth order statistics:

1655699729971.png

1655698794206.png


1655698865555.png
 
Write a computer program (in the language of your choice) that verifies the results in this Example for the case n = 3 and n =4 by taking large numbers of samples.

How to answer this question?
 

Thread 'Formal derivation of statement from Peano Arithmetic system'

I was reading a Bachelor thesis on Peano Arithmetic (PA). PA has the following axioms (not including the induction schema): $$\begin{align} & (A1) ~~~~ \forall x \neg (x + 1 = 0) \nonumber \\ & (A2) ~~~~ \forall xy (x + 1 =y + 1 \to x = y) \nonumber \\ & (A3) ~~~~ \forall x (x + 0 = x) \nonumber \\ & (A4) ~~~~ \forall xy (x + (y +1) = (x + y ) + 1) \nonumber \\ & (A5) ~~~~ \forall x (x \cdot 0 = 0) \nonumber \\ & (A6) ~~~~ \forall xy (x \cdot (y + 1) = (x \cdot y) + x) \nonumber...
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