Cumulative distribution function

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SUMMARY

The discussion focuses on finding the cumulative distribution function (CDF) for 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 emphasize that for a function to qualify as a CDF, it must be monotonic, equal to zero at the lower bound, and equal to one at the upper bound. The conversation reveals that Z cannot take certain values, specifically in the interval (-1, 1), and that the correct approach involves determining the probability density function (PDF) first before integrating to find the CDF.

PREREQUISITES
  • Understanding of cumulative distribution functions (CDFs) and probability density functions (PDFs).
  • Knowledge of integration techniques in probability theory.
  • Familiarity with uniform distributions, specifically uniform random variables on the interval [0, 1].
  • Ability to analyze inequalities and their implications in probability contexts.
NEXT STEPS
  • Study the properties of cumulative distribution functions (CDFs) and probability density functions (PDFs).
  • Learn about the transformation of random variables and how to derive PDFs from CDFs.
  • Explore the concept of joint distributions and independence of random variables.
  • Investigate graphical methods for visualizing the relationships between random variables and their distributions.
USEFUL FOR

Statisticians, data scientists, and students in probability theory who are interested in understanding the behavior of random variables and their distributions, particularly in the context of transformations and cumulative distribution functions.

  • #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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