Cumulative distribution function

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Discussion Overview

The discussion revolves around finding the cumulative distribution function (CDF) of the random variable Z defined as Z = (X + Y) / (X - Y), where X and Y are independent uniform random variables on the interval [0, 1]. Participants explore the properties of CDFs, the limits of integration, and the attainable values of Z.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant attempts to find the CDF of Z but questions the correctness of their integration limits.
  • Another participant emphasizes the requirements for a function to qualify as a CDF, including monotonicity and boundary conditions.
  • Several participants discuss the impossibility of Z taking certain values, such as 0 or negative values, given the constraints on X and Y.
  • There are hints suggesting that a change of variables or finding the probability density function (PDF) might be more effective than directly calculating the CDF.
  • Participants explore the implications of the domain and range of Z, with some arguing about the intervals that Z can take based on the values of X and Y.
  • One participant suggests plotting the function to better understand the range of Z, while others express difficulty in visualizing the function in three dimensions.
  • There is a discussion about the behavior of Z as X approaches certain values, leading to vertical asymptotes and undefined regions.
  • Participants express confusion over the intervals for integration and the correct formulation of the PDF and CDF.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the attainable values of Z or the correct approach to finding the CDF. Multiple competing views and uncertainties remain regarding the limits of integration and the properties of the function.

Contextual Notes

Participants highlight limitations in understanding the mapping from (X, Z) to Y, the need for correct integration intervals, and the challenges in visualizing the function's behavior. There are unresolved mathematical steps and assumptions regarding the values that Z can take.

  • #61
ulissess said:
y>\frac{z-1}{z+1}x==>z>\frac{y+x}{x-y}
In other words, your shaded triangle represents Pr(z>Z). That obviously is not the CDF, but it should also be obvious that it is closely related to the CDF. How?
 
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  • #62
i don't understand very well..
 

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