Finding the Joint and Density Functions for Independent Uniform Random Variables

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SUMMARY

The discussion focuses on finding the joint density function of two independent uniform random variables, X and Y, defined on the interval (0,1). Specifically, it addresses the transformation to new variables U=X and V=X+Y. The joint density function is established, and the density function for V is computed using specified bounds. The bounds for V are clarified as 0 PREREQUISITES

  • Understanding of joint probability density functions
  • Knowledge of transformations of random variables
  • Familiarity with integration techniques
  • Concept of independent random variables
NEXT STEPS
  • Study the derivation of joint density functions for transformations of random variables
  • Learn about the properties of uniform distributions and their applications
  • Explore the concept of convolution for finding the sum of independent random variables
  • Investigate the use of integration bounds in probability density functions
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Students and professionals in statistics, probability theory, and data science who are working with random variables and need to understand joint distributions and density functions.

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Homework Statement


Let X and Y be independent uniform (0,1) random variables.

a. find th ejoint density of U=X, V=X+Y.

b. compute the density funciton of V.

Homework Equations





The Attempt at a Solution



Part a. is not a problem. I don't understand how the bounds for part b. are set up. The book says: for 0<V<1, fv(v)=\int_{o}^{v}du
and for 1\leqv\leq 2: \int_{v-1}^{1}du

Could someone explain this to me?
 
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Could somebody explain this in English?
 

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